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.
Problem?
In this particular case, nothing. Any concern you might have about "overpowered" is mitigated by his stats being at unimpressive levels. From your previous question you indicate that the players are not leaning toward min-max, power gaming, nor optimization in a munchkinny way. He's using finesse weapons that in time may limit highest damage potential: but if they aren't in min max mode, that does not seem to matter to the group.
At higher levels, he may get a little frustrated with melee damage since there are no two-handed finesse weapons and there are fewer choices in weapons ... but a lot of that gets into what fighting styles he uses and chooses. (And you control what magical weapons, if any, become available so you can tailor that to this party/team. Our DM did just that for our Bard, who is a rapier using singer).
At higher levels he'll benefit from those Divine Smite nova shots. When paired with sneak attacks from his rogue levels, he should be able to keep his damage in a "good enough" zone if the party fights well as a team.
The other place a low Strength would be a Paladin issue is in armor selection: if he wants Chain, Splint or Plate, he needs 13 / 15 / 15 strength respectively. With the proposed Dex build and Rogue multiclass, he's unlikely to want them since they give Disadvantage on Stealth checks. He can wear Breastplate with no minimum Strength requirement -- allows up to +2 of his Dex AC bonus, and no disadvantage for stealth. At 400 GP it may be a level or two before he can afford it. A 50 GP chain shirt may be his best for the time being.
From @Ethan's comment: With leather armor, a shield, and 20 Dex (After a ASI's) he will end up with 19 AC, 1 lower than full plate+shield, but the Piradin will be good at all +Dex skills. This makes for solid function when full plate isn't a good idea -- such as waterborne adventures. His damage output will be identical to a strength based shield using paladin (barring difference in MC class features).
If this is a role and story based move, all the more reason to go with it. It sounds like fun - which is why we play. As long as the player knows where the difficulties are up front with the stats as is, the player is taking on a challenge. Good on 'em!
I'd recommend a ruling to let him proceed as desired. (And if later on they finds some gauntlets of ogre strength ... )
Best Answer
Your Proficiency Bonus is based on the total levels you have as a character, not on any one class.
The proficiency bonuses listed in the Ch. 1 table are the same as those listed in each class's advancement table. That is, the bonuses are identical across classes, so there's no differentiation between the "fighter 13 bonus" you mention and the "[anyclass] 13 bonus" or the "[class1] X/[class2] 13-X bonus".