This is the Reti Opening, named after Richard Reti, an Austro-Hungarian chess master.

This can lead to almost anything: Reti, King's Indian Attack, or transpositions into more usual lines.

1...Nf6 is a fine response, as non-committal as White's opening move. I would not recommend 1...e5, as White can play 2.Nxe5, winning the pawn - a centre one, be it noted - at a small cost in time.

Perhaps this is a little too 'non-committal'. I would suggest here 2.g3, or 2.d4 or 2.c4 as very good moves.

2...g6, or 2...e6, or 2...d5 or 2...c5 were all good replies to White's 2nd move.

'Up' = 'advanced', then. This is conservative, but fine. More active would have been 3.e4, with a 'Four Knights' game in mind.

Not really a 'trap' as White might well be quite happy with the 'equal trade.' (by which I infer Black has ...d4 in mind). This move is quite OK, leading to a QP game after 4.d4.

This knight sortie is ill-advised, as White has no real targets at which to strike. Black will be more than happy for White to take (5.Nxf7 Kxf7) as a pawn plus the slight disruption of the Black king position is a poor bargain for a whole piece (knight).

This Queen move is premature, and could have lost Black time after 5.Nb5!. Note that this Knight sortie would have been justified by the attack on the Black Queen and the c7-square as well. Black would pretty much have had to retract his Q move altogether (5...Qd8), which would have compensated White for the time he has lost with his Ng5 move. There is sometimes a very fine line between what makes a move good or bad. It may simply be a matter of timing.

I agree with isaiah11: the best move was 5...Qd8, admitting the mistake.

I think I might have taken with the other pawn, but there's little in it. Black has very little for the Queen: a knight and a slight lead in development - not nearly enough. But if Black plans to play on, she should be prepared to play very actuively, and look for threats. First priority, though, is to complete development: ...h3 (to drive back the knight) and ...0-0-0.

doubled pawns here are fine - a minor inconvenience only, and both sets go to strengthening the centre. But the issue is very minor in the present circumstance. Black needs to mobilize quickly, and that (more or less) justified accepting the exchange White offered.

Too weakening, and doesn't go towards mobilizing. Better was 8...h6, to drive away the knight, allowing Black then to play 9...0-0-0.

Tripled centre pawns! But not for long.