The DM needs to adjudicate portions of this.
What kind of action is it to pick it up? Does it take a whole action or can it be done as a "free object interaction"?
It uses your full action. The DM will likely call for a Strength (Athletics) check to carry the body. PHB pg. 190 has examples of things you can do with your "free" interaction, and nothing even begins to approach moving a body. In fact, it takes an action just to put on a shield that you're already carrying, or to take out a second weapon.
Can the PC move at full speed while carrying the body or are there any speed penalties?
Grapple rules aptly apply to this. The difference between an unconscious person and a dead one is a pulse. Effectively, apply grapple rules. So half speed at best. More if you lack Strength or if your DM is adhering to encumbrance rules.
Does the PC have any limitations to other actions while carrying the body? Can the PC carry the body on his shoulder so he still has both hands free to attack, etc.? Does the PC get disadvantage on attack rolls or anything while doing that?
Again, I would apply grapple rules since you're holding something. That means at least one hand is occupied. As for the rest, if you are Encumbered apply encumbrance rules. These can be found in the PHB pg. 176; Variant Encumbrance.
The rules for lifting and carrying are intentionally
simple. Here is a variant if you are looking for more
detailed rules for determining how a character is
hindered by the weight o f equipment. When you use this
variant, ignore the Strength column of the Armor table
in chapter 5.
If you carry weight in excess of 5 times your Strength
score, you are encumbered, which means your speed
drops by 10 feet.
If you carry weight in excess of 10 times your
Strength score, up to your maximum carrying capacity,
you are instead heavily encumbered, which means your
speed drops by 20 feet and you have disadvantage on
ability checks, attack rolls, and saving throws that use
Strength, Dexterity, or Constitution.
As stated above, the rules are intentionally simple. You describe what you're doing, the DM tells you what he wants you to roll. That's basically the bottom line here.