To the first part of your question, you don't gain the bonus feat progression.
This ability does not automatically grant feats normally granted to fighters and monks based on class level, namely Stunning Fist.
As a 5th level Brawler, you'd add your Brawler level (5) to your effective monk level (0) and effective fighter level (0) for the purposes of feat prerequisites and other interactions.
For the second part,
While a character can multiclass with [a hybrid class's] parent classes, this usually results in redundant abilities. Such abilities don't stack unless specified.
I'm not entirely certain how this interacts with damage dice, but a ruling in line with intent is that you use the larger of the two damage dice. A fair ruling could allow the progressions to stack anyway.
AC bonus is questionable, due to the abilities sharing the same name. I personally would rule that they do stack, due to different typing.
This is one of the most annoying and awkward pieces of 3.x rules
You don’t get enhancement bonuses to AC, you get enhancement bonuses to armor bonuses
So, armors grant an armor bonus to AC. Seems simple enough.
Then we have enhancement bonuses. You could, in theory, have an enhancement bonus to AC; if you did, it would stack with an armor bonus, since they are different bonuses. However, as far as I know, there is no effect in the system that gives a creature an enhancement bonus to AC.
No, instead enhancement bonuses are applied to items. When you wear magically-enhanced armor, you don’t have an enhancement bonus, the armor has an enhancement bonus – to the armor bonus that it gives you. So in short, your AC doesn’t “see” an enhancement bonus at all; it just sees a (larger) armor bonus.
A similar example exists in the amulet of natural armor’s “enhancement bonus to your natural armor bonus.” This time, the enhancement is on the person wearing the amulet, but it’s not an enhancement bonus to his AC, but an enhancement bonus to his natural armor bonus. As far as the wearer’s AC is concerned, it sees no enhancement bonus, just a larger natural armor bonus.
And so it is with magic vestment: it applies an enhancement bonus to an item’s armor bonus (including an armor bonus of +0 from a non-armor). Your AC still only sees a larger armor bonus, not any enhancement bonus.
Thus you might have a shirt that has a base armor bonus of +0, to which it receives a +4 enhancement bonus. The wearer of that shirt gets a +4 armor bonus. If they are already wearing full-plate, the +8 armor bonus of that supersedes the +4 armor bonus of the shirt, so the shirt becomes pointless. On the other hand, magic vestment could be cast on the full-plate, giving it a +4 enhancement bonus, improving its armor bonus to +12. If it had instead been a +1 full-plate, the +4 enhancement bonus would supersede, not stack with, the +1 enhancement bonus, so the armor bonus to the wearer’s AC would still be +12.
Best Answer
1) It is a coat. The line "You can use this coat" covers that for me. 2) and 3) Its possible that it falls in between, but see below.
Common terms states:
Heavens Coat of Many Stars states:
Conjuration Creation (which is just for completenes, since the effect isn't called out as any specific school) states:
From this we can determine that you are in fact wearing an item of clothing, its not just shiny lights, it is in fact an object (though don't ask me how you sunder it), and furthermore, it is specifically called out as clothing (coat).
Magic Vestment states:
I would argue that "regular" in this case means "not armor", rather than "commonplace". I'm pretty sure it was not intended to a silken shirt could not have magic vestment cast upon it.
Now, either the coat is armor (because it provides an armor bonus) or it is not (and is therefore clothing). Either way the spell works.
Note: A really stingy DM could argue that clothing doesn't have an AC bonus and therefore the second part of this doesn't apply, in which case your stuck with your DMs interpretation. However, since the spell goes out of its way to basically say "armor or just whatever your wearing" I think the fact that is intended that it should include special cases that fall in between. Point out that Oracles can normally wear armor anyway, and that Mage Armor, Magic Vestment and normal clothing would work.
( All emphasis mine )