This is not the modern (which is 1...g6), but Owen's Defence.

Let's say it is 'a' modern opening. Even less than 50 years ago, moves like this tended to attract sideways looks from chess commentators. Things are a deal more liberal now...

Not the best diagonal for the bishop though, as it is biting on granite. You can also chase it with ...d5 sometime, but that has the disadvantage of blocking in the light-squared bishop if it has developed to b7.

Doesn't black lose material with 3...d5 because white can take with the e-pawn and has the bishop and the knight to protect it? Unless you mean to play d5 after you play Bb7

Your last move was a mistake. White could have proved that by playing 4.e5. The knight has to move again and 4...Nd5? loses a pawn (d5 is attacked twice and defended once), while 4...Nh5??, 4...Ne4?? and 4...Ng4 all lose the knight immediately. So the only safe square for the knight is on g8.

After 4.e5 d5, black is OK. Most likely the opening transposes to a French defense advance variation

I would prefer 4...Bb7.

4...d5 is better than 4...Bb7.

Both are OK and thematic.

Now after 5.e5 5...Nfd7 is possible, which is no problem at all because the bishop wants to go to b7 anyway. But I would play 5...dxc4 6.exf6 Qxf6 7.dxc4, gaining development and the bishop pair.

As I said before, the bishop should go to b7, so this is surely the best bishop move. I don't know if 6...Qe7 would be better though.

Depends if you like 6...Qe7 7.Nxd5 Nxd5 8.Bxd5 for Black...

Bad move, you can easily counter this and he moves the same piece twice in the opening.

He had to move the bishop anyway, but Bb3 seems more circumspect.

Did I say twice? ... Well, that should be thrice.

Black has now equalized.

Nice! Taking advantage of the pinned bishop.