I am trying to make a homebrew slime race, and I'd like to know what complications might arise from a PC having the Ooze type, including ones caused by having Ooze AND Humanoid.
On its own, the ooze type has (almost) no direct impact
There are virtually no race/class features or feats (or spells, or magic items) that apply specifically to oozes.
Undead and oozes have disadvantage on this saving throw.
The ooze creature type can also be chosen by a ranger as a Favored Enemy, but monsters/NPCs aren't generally built like player characters are, so that's not really a concern.
If the playable race has both the humanoid and ooze types, and we assume that things that apply to humanoids still apply to them, then almost nothing really changes based on just that.
...But there may be other implications
The Monster Manual section on oozes (p. 240) states:
Oozes thrive in the dark, shunning areas of bright light and extreme temperatures. They flow through the damp underground, feeding on any creature or object that can be dissolved, slinking along the ground, dripping from walls and ceilings, spreading across the edges of underground pools, and squeezing through cracks. The first warning an adventurer receives of an ooze’s presence is often the searing pain of its acidic touch.
Oozes are drawn to movement and warmth. Organic material nourishes them, and when prey is scarce they feed on grime, fungus, and offal. Veteran explorers know that an immaculately clean passageway is a likely sign that an ooze lairs nearby.
So rather than just eating "normal" food, they can subsist on the remains of basically any organic material.
Slow Death. An ooze kills its prey slowly. Some varieties, such as black puddings and gelatinous cubes, engulf creatures to prevent escape. The only upside of this torturous death is that a victim’s comrades can come to the rescue before it is too late.
Since not every ooze digests every type of substance, some have coins, metal gear, bones, and other debris suspended within their quivering bodies. A slain ooze can be a rich source of treasure for its killers.
The ability to basically subsume anything into its body by engulfing it can be a little too strong, though it'd only apply to creatures of its size or smaller. Still, potentially an overpowered ability at early levels.
Unwitting Servants. Although an ooze lacks the intelligence to ally itself with other creatures, others that understand an ooze’s need to feed might lure it into a location where it can be of use to them. Clever monsters keep oozes around to defend passageways or consume refuse. Likewise, an ooze can be enticed into a pit trap, where its captors feed it often enough to prevent it from coming after them. Crafty creatures place torches and flaming braziers in strategic areas to dissuade an ooze from leaving a particular tunnel or room.
Often, oozes are not very smart. This could simply be manifested as a racial penalty to one's Int score, though it's not necessarily true of every ooze (the oblexes from Mordenkainen's Tome of Foes are particularly intelligent, especially the Elder Oblex with its Int score of 22).
Spawn of Juiblex. According to the Demonomicon of Iggwilv and other sources, oozes are scattered fragments or offspring of the demon lord Juiblex. Whether this is true or not, the Faceless Lord is one of the few beings that can control oozes and imbue them with a modicum of intelligence. Most of the time, oozes have no sense of tactics or self-preservation. They are direct and predictable, attacking and eating without cunning. Under the control of Juiblex, they exhibit glimmers of sentience and malevolent intent.
This may really be the main concern - oozes are often thought to be connected to the chaotic evil demon lord Juiblex. (The other main concern being that oozes are generally considered weird and foreign, so it might be tricky getting any humanoid to trust them or work with them...)
Ooze Nature. An ooze doesn’t require sleep.
Oozes don't generally need sleep. This is the easiest to work with, as there are other playable races that don't need sleep (elves can just take a 4-hour trance instead, and warforged don't need to sleep at all). This trait also seems the least essential to the flavor of an ooze race.
It's also worth looking at the Slithering Tracker, the only ooze in Volo's Guide to Monsters (p. 191):
The quest for revenge sometimes leads one to undergo a ritual whereby they transform into a body of semiliquid sentience known as a slithering tracker. Innocuous and insidious at the same time, a tracker flows into places where a normal creature can’t go and brings its own brand of watery death down upon its quarry.
Vengeance at Any Cost. The ritual for creating a slithering tracker is known to hags, liches, and priests who worship gods of vengeance. It can only be performed on a willing creature that hungers for revenge. The ritual sucks all the moisture from the person’s body, killing it. Yet the mind lives on in the puddle of liquid that issues forth from the remains, and so too does the subject’s insatiable need for retribution.
Stealthy Assassins. A slithering tracker tastes the ground it courses over, seeking any trace of its prey. To kill, a slithering tracker rises up and enshrouds a creature, attempting to drown the prey while also draining it of blood. A slithering tracker that has killed in this fashion becomes much easier to locate for a time, since its liquid form becomes tinged with blood and its body leaves a visible trail of the stuff behind it.
Descent into Madness. Achieving revenge against its target doesn’t end a slithering tracker’s existence, nor its hunger for blood. Some slithering trackers remain aware of their purpose and extend their quest for vengeance to others, such as anyone who supported or befriended the original target. Most of the time, though, a tracker’s mind can’t cope with being trapped in liquid form, unable to communicate, and driven by the desire for blood: after a tracker fulfills its duty, insanity takes over the creature, and it attacks indiscriminately until it is destroyed.
Probably not the best template for a playable race, but might provide some helpful ideas.
And finally, the oblexes in Mordenkainen's Tome of Foes (p. 217):
By experimenting on the slimes, jellies, and puddings that infest the depths of the Underdark, mind flayers created a special breed of ooze, the oblex — a slime capable of assaulting the minds of other creatures. Cunning hunters, these pools of jelly stalk prey, searching for the memories they so desperately crave. When oblexes feed on those thoughts, sometimes killing their victims, they can form weird copies of their prey, which help them to harvest even more victims for their dark masters.
Memory Eaters. Oblexes feed on thoughts and memories. The sharper the mind, the better the meal, so oblexes hunt obviously intelligent targets such as wizards and other spellcasters. When suitable fare comes within reach, an oblex draws its body up to engulf its victim. As it withdraws, it plunders the creature’s mind, leaving its prey befuddled and confused.
Ooze Nature. An oblex doesn’t require sleep.
Setting aside the traits, this might be the best inspiration for how an ooze playable race came to be - though you'd have to consider why/how they're not evil. Oblexes engulf their victims, raid its mind and memories, and then are able to impersonate that target.
Mordenkainen's Tome of Foes Chapter 1 (p. 30) has a section on demonic boons that a DM can grant to monsters and NPCs dedicated to a particular demon lord, which you could use as inspiration for racial traits as well:
Wicked folk who seek power from demons are scattered across the multiverse. Some of them gather in cults, but many of them act on their own or in small groups. Whatever their organization, they are united in their desire to draw power from the bottomless evil of the Abyss.
The following entries outline boons that a DM can grant to monsters and NPCs dedicated to a particular demon lord. The entries also list signature spells associated with a demon lord. If the monster or NPC can cast spells, you can replace any of those spells with spells from that list, as long as the new spell is of the same level as the spell it replaces.
Ability Score Adjustment: Up to a +8 bonus to Constitution, with an equal penalty to Intelligence, Wisdom, and Charisma
Signature Spells: Grease (1st level), web (2nd level), gaseous form (3rd level)
The Faceless Lord’s followers are bizarre loners who prefer the company of slimes and oozes to other creatures. They gain the hardiness of a slime at the cost of their minds. Lesser followers gain the Liquid Movement trait. The most dedicated devotees of ooze also gain the Slimy Organs trait.
Liquid Movement. As an action, this creature can move up to 20 feet through spaces no more than an inch in diameter. It must end this movement in a space that can accommodate its full size. Otherwise, it takes 5 force damage and returns to the space where it began this movement.
Slimy Organs. This creature has resistance to bludgeoning, piercing, and slashing damage from nonmagical attacks.
Whenever this creature suffers a critical hit or is reduced to 0 hit points, all creatures within 5 feet of it take acid damage equal to its number of Hit Dice.
Obviously, you should keep balance in mind when considering whether to grant any of these benefits, and how to balance any added benefits against other races (perhaps by granting other downsides). The ability score adjustment would likely be unbalanced; the signature spells could be allowed if you follow the format of other racial spellcasting traits; Liquid Movement could be allowed as long as you design/modify the campaign with it in mind; and Slimy Organs would likely be overpower.
All in all, you'd likely want to avoid giving many of these traits/abilities to a playable race, but they are things you'd want to keep in mind as you make an ooze playable race. You can, of course, avoid incorporating most of these traits, especially if your campaign is set somewhere other than in the Forgotten Realms (which the Juiblex connection is tied to).