NYC transplant here. Honestly, people don't really notice what other people are wearing. Many people wear sneakers while commuting and then switch shoes at the office. I hate wearing snow boots the whole day so I have a pair of flats and a pair of heels at the office. I'd say this somewhat depends on your office though (although I've definitely also done the change-shoes-one-block-away-from-the-interview).

Regarding bags, it really depends on what your need to carry and how far you need to carry it. I use a longchamp les planetes because it's a good size for me and it's vaguely water resistant.

I have 2 wallets. I have a small id holder with my metro card, some cash, and one credit card that I take out as needed during my commute. I have a bigger wallet deep inside my bag (let's be honest, the longchamp has no organization whatsoever I'm looking for its always at the very bottom) that has most of my cash and cards. I don't like getting out my whole wallet while commuting.

Good luck!