It's important to find the fun for your table before putting much effort into mechanics
A key observation in this case is that D&D 5e doesn't simulate much around causing or spreading fires in combat, which matters because there aren't very many mechanics around those things. As you've noted, the core rulebooks generally only go as far as indicating that an object could be lit on fire if it's flammable.
That's the first big consideration: detailed mechanics around how things burn (how easily they ignite, how quickly they are destroyed by fire, how fire might naturally spread around the area) don't really exist, and so neither do applications of those properties. So before working out a detailed system to apply those properties it's worth considering why it will be worth your (and your players') attention to interact with them. And with homebrewed mechanics that can get really table-specific and so it's a great opportunity to create something tailored to your players.
For example, if your players have fun preparing battlefields with hazards and like the idea of preparing burn zones mid-combat, then details like how easily or how quickly fire spreads from one object to another would be important to focus on and how much damage an object takes from being on fire may be less so. Then it's more a question about what types of objects PCs might be interested in moving around and burning and less about how durable they are to flames.
My players definitely like getting extra mileage out of their spells, including starting fires, but they are not so interested in very precise object interactions like burnability of random objects. So for my players it would be too fussy and niche-applicable to worry about those kinds of details, and it would degrade the game to spend time modeling and considering them all the time. We still do all kinds of fire-related stuff, but I fold it into the streamlined nature of 5e: things ignite, or don't, producing environmental hazards; interacting with burning objects grants Advantage (like when trying to break down a door) or Disadvantage (like when trying to do investigate the interior of a building that's on fire); fire damage translates directly to what a fire from Create Bonfire would do; and other things like that.
But that's us, and other tables might not have as strong a preference for ease and fluidity of play versus more number-crunching content.
I do not recommend developing and choosing mechanics in a vacuum, then seeing what happens; I recommend thinking about what you want to be possible (but currently isn't) and then create new mechanics to make it possible.
In games that I run, my players try to get all kinds of extra benefits out of spell effects. That very much includes fire and heat. But what I have discovered my players most want are, in roughly this order, to:
- Think of clever solutions to problems that use their characters'
- Overcome in-game obstacles more easily than what they assume the
intended approaches would allow
- Do things with some extra dramatic flair
I'll give two examples of fire spells in this context from a game I'm running now.
In the first case, the PCs were sneaking into a building to steal some sensitive documents and discovered a prisoner caged in the office they were ransacking. My most fire-happy player decided that they should steal the documents and free the prisoner, then use a fire to destroy the evidence. I ruled that it worked, but that the fire quickly spread out of control and engulfed the entire building. The flammability of the building created a narrative situation in which they had an effective time limit to escape alive, and with certain routes blocked off due to a new environmental hazard. Plus lots of complications after the fact.
In this case, more specificity wouldn't have made the encounter more interesting but definitely would have multiplied the amount of work I had to do. Tracking the flammability of the wooden desk vs. the wooden floor vs. the bookshelves would have added a lot of bookkeeping, and what I would have gained from it would have been a more slowly-spreading fire (and therefore fewer complications to the encounter) and a more formulaically-spreading fire. More predictability and less peril is not what I was going for. Even if I had had house rules for flammability of all objects in place, I would have thrown them out for this sequence.
What I ended up choosing interacted with elements 1 and 3 from my list: fire magic made hiding evidence easier, and was certainly dramatic. But while the player expected to get 2 as well (no evidence, we'll be totally in the clear!), they got the opposite. Which also made sense in-game, as that character had no expertise in forensic fire investigation and should not be able to casually predict exactly how a flammable building will ignite.
The same party in the same game came upon a locked vault door with very little time to break in but a strong interest in doing so. They had not prepared enough to get into the vault by any means I had planned, and a vault door that can be casually bypassed at any time may as well not be there at all. They learned through investigation that the most vulnerable part of the vault door was a collection of hinge mechanisms that would be very hard to destroy in the time they had available. A player came up with the idea of using magical fire to heat the hinge mechanisms and make them easier to break.
That's not too realistic, but I ultimately said yes: PCs attacking a hinge mechanism in the middle of a bonfire got to attack with Advantage. It was a novel use of the PC's magical abilities (item 1), helped them deal with an obstacle in a novel way (item 2, and especially important as they had no other way of getting in), and was dramatic (item 3). It doesn't matter that thick metal objects aren't flammable, or that I didn't have a full suite of mechanics dealing with heating metals. The party got to solve their immediate problem with creative use of the resources at their disposal. The mechanics of how that played out were the least important part of what was a tense and fun encounter.
Regarding the specific questions posted in the OP:
- Should flammable objects be vulnerable to fire damage?
I say that this one is largely irrelevant. Objects being destroyed or not by the end of combat usually doesn't matter at all, and consequently neither does how quickly or easily an object is destroyed. In cases where this does matter (and you could definitely engineer as many of these as you like), the challenge is almost certainly going to come from issues that surround the object's destruction. Those are in your hands as DM, and creating the scenario will require you to account for this one way or another.
As an example, if your players need to destroy a flammable object within 10 turns to accomplish their objective, and vulnerability to fire would make that too easy to do, then you would either need to increase the object's HP, introduce other obstacles which prevent the PCs from igniting it too early in the combat (or something similar), or choose to use a non-flammable object instead. Any such mitigation would just make the vulnerability irrelevant. So this idea, to me, just feels like mechanic and bookkeeping bloat.
- Should fire-damage spell attack rolls have advantage against
As with the previous bullet point, I don't think that it matters. But if pressed to it I'd say no. An attack roll for a fire spell is usually to see if the spell itself hits in a meaningful way, not to see if the target ignites. If you miss with Firebolt, the bolt of fire doesn't hit what you were trying to hit. Being flammable alone doesn't make a target easier to hit, only easier to set ablaze if it is hit.
If you have flammability as an option with meaningful gameplay benefits and a fire-happy table of players, you are already in for a lot of fires. Granting Advantage to efforts to start fires will exacerbate this while also making those benefits easier to gain. But if you're ready for those, then I don't think that this one would matter much either way as it would be subsumed by encounter design concerns.
- How should the ongoing damage of something being 'on fire' be
I, personally, just treat burning objects as though they were in the middle of a fire created by the most basic casting of the Create Bonfire spell but without the opportunity for a Saving Throw. If something is on fire, it can't dodge the heat or flames! So it's generally 1d8 per turn.
- Is the descriptor 'flammable' applied to objects at the DM's
Of course. The word flammable itself is a kind-of-subjective term: many things will burn in the right conditions, but most people are concerned with the conditions they are most likely to encounter. Water isn't generally thought of as flammable, but hurl a chunk of cesium into it and you'll see it "burn". Wax candles are designed to burn in everyday conditions, but a candle in a vacuum will never ignite in candles' usual way.
The importance of whether or not an object is described as flammable can only be evaluated in the context of how important you have decided a burning object will be. But I do not think that a DM adding or removing that description from objects will, in itself, have much material effect on the game.