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Is is better to have a weak and doubled extra pawn or to allow your opponent to exchange and keep a solid structure?
White needs to defend his d-pawn with Be3 or c3. If black decides on Bxf3, then gxf3. Should white decide upon c3, after exd4, black has an annoying check with Bb4. On the other hand, white has 2 center pawns and black has none.
White needs to make sure his center pawns are defended and not put anything inbetween his queen and his d-pawn.
Don't capture the e pawn.
As well as what has already been said black has two good moves if you do:
...Qxd1; Kxd1,Rd8 and the White King cannot castle and has blocked his Rook in.... or
Qxd1; Kxd1,Bxf3;gxf3... and white now has two sets of doubled pawns (although he does have a door for his rook.
...Qxd1; Kxd1,0-0-0 is best for black
This is a line out of the exchange ruy. (More normal is for white to castle on move 5). Here the book move is to capture black's pawn on e5. This is not a weak doubled pawn.
There is a similar position which may be better for your question (when white does castle on move 5).
And the way you say "or allow your opponent to exchange and keep a solid structure" it sounds like you mean if you take on e5 he won't have a solid structure... black will still be solid, but his doubled c pawn isn't ideal, i.e. his structure has problems either way (the benefit is his queen and bishop were opened, again the sacrifice of structure for activity).
The question is, if taking on e5 results in less active / well placed pieces. Black may or may not gain back the pawn, but what you count as a success in the opening is efficient development not a pawn grab.
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