For clothes, check out SoHo. Broadway in particular is lined with big name brand stores (Gap, Banana Republic, Bloomingdales). But the side streets (Mercer, Wooster, Greene) are full of tiny high-end boutique shops (and not just for clothing).

There's also a smaller strip of name brand shops on 5th Avenue just north of Union Square (around 18th St). This is extremely close to your hotel, just a 5-minute walk.

For department stores, check out Macy's Herald Square or Century 21. The Macy's is the company's flagship location and a major New York institution (ten floors of everything). It's one of New York's shopping "must-sees". Century 21 is a much beloved discount department store where great deals can be found in return for chaotic crowds that swarm the place. Best to visit early and on weekdays.

For an iPod, definitely go to an Apple Store. The Apple Store Fifth Avenue is a flagship store at the end of the 5th Avenue shopping district (5th Ave & 59th St, at the corner of Central Park). You can also stop by the Apple Store SoHo (103 Prince St, two blocks west of Broadway), which is convenient when you're already in SoHo.

I can't say I'd recommend anywhere else. There are your usual big box stores (Best Buy, including one across from your hotel), but they're nothing special. Stay away from the small electronics shops that you'll find all over Midtown.

I can offer some more suggestions if you like. There are a few other good spots, depending on what you're looking for.

SoHo runs between Houston St (to the north) and Canal St (to the south). It's about 20 blocks south of your hotel. You could walk there in 20+ minutes or take a 5-minute subway or taxi ride.

Century 21 is in the Financial District (lower tip of Manhattan), by Wall Street and the World Trade Center. This is farther away, too far to walk.

I put up a bunch more detail on these places on your guide. That'll have all the info you need.

Given the way your schedule is going, I'd say Tuesday, after the Statue of Liberty, or Saturday (New Year's Day). If you want you can fit this into any day, but it'll probably be easier when you're running around everywhere in Midtown.

Oh, absolutely. Especially all the big shops. Some of the smaller boutiques might be closed, but you'll have no problems at all. Plenty of shopping goes on on New Year's. You'll barely notice any difference from any other day.