Yes...go! I've tagged along on DH's business trips with my little ones plenty of times. I even did the St. Patrick's Day parade alone when it was dd2 and dd 6weeks!!
NYC is amazing at Christmas! As for pace...just go your own pace. Here are some ideas:
FAO Schwartz
Lord and Taylor Windows
Toys R Us
Disney Store
Santa Land at Macy's (yes!...yes!)- We went all the way to NYC for dd1's first visit to see Santa to do it here. They had a puppet show too!
The Rockettes can be pricey and a three year old may or may not make it through the whole show. You know your child best to judge this!
When all else fails, just head to the Manhattan Children's Museum on the Upper West Side. It's warm, its child-friendly, and it's fun! (and lunch at Artie's Deli just around the corner afterwards is great too!)
I do take my children (now 5 and 2) on the subway...but I didn't when they were really little because I couldn't manage the big stroller and carry the baby at the same time. If you are just there for a short visit, just cab it....it is a bit pricey, but so worth it. You can get cabs at your hotel and at Macy's cab stand really easily....(and just put out your hand, of course).
To be honest, with dd5 and dd2, I don't really get all worked up about the car seat for the short rides we take..I just strap them into using the seat belts. I'm sure there are some purists who would be appauled, but it works for me and for the cab drivers! I'm sure you can always call a car service to assure you have access to a car seat, but I would just strap a three-year old in using the seatbelts.
The city is just magical....just walking around with dd3 will be a blast!