Nous flânons à la brocante et Paul marchande.

Because the subjects are different.

• nous = we, so the verb takes the 1st person plural ending
  
  • flâner → nous flânons
• Paul = he, so the verb takes the 3rd person singular ending
  
  • marchander → Paul marchande

Both verbs are in the present tense.

So the pattern is:

• nous flânons = we are strolling / we stroll
• Paul marchande = Paul is bargaining / Paul bargains

Here, marchande is a verb, not a noun or adjective.

• Paul marchande comes from the verb marchander = to haggle, bargain
• It means Paul is bargaining / haggling

This is different from the noun:

• un marchand = a male shopkeeper / merchant
• une marchande = a female shopkeeper / merchant

So in this sentence, the -e has nothing to do with gender. It is simply the normal present-tense ending of the verb for il/elle/on and for a singular name like Paul.

Not exactly. Flâner is more specific than just to walk.

It usually means something like:

• to stroll
• to wander around leisurely
• to amble
• to browse while walking around

It suggests moving slowly, without hurry, just enjoying the place.

So:

• marcher = to walk
• se promener = to go for a walk / take a stroll
• flâner = to stroll around aimlessly or pleasantly, often looking at things

In this sentence, Nous flânons à la brocante gives the idea that we’re wandering around the flea market, looking around casually.

Because the infinitive is flâner, and that word is always spelled with â.

So you get:

• flâner
• je flâne
• nous flânons

For most learners, the main thing is simply: memorize the spelling as part of the word.

The circumflex does not change the basic grammar. It is just part of the standard written form of the verb. In pronunciation, it can slightly affect the vowel quality, but for beginners the important point is that it belongs to the word.

À la brocante means at the flea market / at the second-hand market.

French often uses à for being at a place, event, or destination:

• au marché = at the market
• à la plage = at the beach
• au restaurant = at the restaurant
• à la brocante = at the flea market

Here:

• à = at / to
• la = the
• brocante = flea market, second-hand sale, antiques/bric-a-brac market

Why not dans la brocante?
Because à la brocante is the more natural, idiomatic way to say at the flea market as a location or event. Dans would sound more like inside something physical.

They are very similar, and in many situations they can be translated the same way: flea market.

But there can be a slight nuance:

• brocante often suggests second-hand items, old objects, antiques, bric-a-brac
• marché aux puces is a very common term for a flea market

Depending on region and context, brocante may sound a little more like a sale or market focused on old or used objects, sometimes a bit more charming or antique-oriented.

So in this sentence, à la brocante is very naturally understood as at the flea market / second-hand market.

In everyday spoken French, many speakers would more naturally use on instead of nous:

• On flâne à la brocante et Paul marchande.

That sounds very natural in conversation.

However, nous flânons is still completely correct. It may sound:

• a bit more careful
• a bit more written
• slightly more formal or literary

So the sentence is not wrong at all. It is just a little less conversational than the version with on.

A simple pronunciation guide is:

Nous flânons à la brocante et Paul marchande.
noo fla-non a la bro-kont ay pol mar-shond

A few useful points:

• nous → the final s is silent
• flânons → the final -ons has a nasal sound
• à la → both vowels are pronounced clearly
• brocante → the final e is silent
• et → pronounced ay
• Paul → sounds roughly like pole
• marchande → the final e is silent; the an is nasal

A more IPA-style version is:

/nu flɑ.nɔ̃ a la bʁɔ.kɑ̃t e pɔl maʁ.ʃɑ̃d/

Because the sentence has two coordinated clauses joined by et:

• Nous flânons à la brocante
• et Paul marchande

The second clause has its own subject, Paul, so there is no reason to repeat nous.

In English, this works the same way:

• We stroll around the flea market and Paul bargains.

French is doing the same thing here: one subject for the first verb, a different subject for the second verb.