I hike all throughout the NE in the winter months. I will try to answer all our questions the best I can.
Sleeping bag: At the very minimum you should take a quality 0F bag. But a -15F to -30F bag are recommended. You can easily extend the range of your bag with clothing, bivy, tent etc. But you may be in for some chilly nights with just a 0F bag. That being said, I use a 0F top quilt with capaline 2 long john bottoms, cabelas down pants, down booties, capaline 3 top and a nano puff and down vest down to -30F with no issues. As always the hot water in a nalgene is an awesome trick.
Boots: This is a widely personal question and you will recieve lots of different opinions regarding such. I use a pair of TNF Arctic pull ons, and smartwool mountaineering socks while hiking/snowshoeing and am perfectly fine. In camp I use my down booties typically, but with my boots and dry socks I am comfortable to -15F or so. I really like pull ons for winter because they wont freeze to the point where you can't put them back on. And it makes for easy on and off with gloves etc.
Pants: While hiking I typically just use a pair of winter weight BDU pants, if its really cold I wear a pair of capaline 2 long johns. I also have ECWCS goretex pants as a hardshell as needed. In camp I wear cabelas down pants under the bdu pants.
Tent: I use a hammock, tarp with doors, top and under quilts, and a winter hammock sock. Not your norm by any means. Bottom line is you need a solid shelter and not rely on building a snow shelter especially for the first time without practice. If your gonna be basecamping feel free to give it a shot, but its not easy the first time. Even if you do plan on doing a snow shelter, at a bare minimum bring a tarp.
Crampons: This depends. If your going to be going on very steep slopes then yes real crampons and ice axe are needed in alot of places. However, i have been able to get by with microspikes and msr denali ascent snow shoes. The snowshoes crampons can be removed and used seperately, which work ok but arnt near as nice as real crampons. I typically dont try to summit peaks in the winter, simply because I have a dog with me and am solo 99% of the time.
Poles: you do not 'need' snow baskets, but they do come in handy otherwise your poles will sink very far and can be near useless at times. Snow baskets are cheap, alot of brands will fit any pole. You can also make your own with a little ingenuity and some stiffish plastic like milk carton or 2 liters cut up. Old plastic plates etc.
Snow shovel: yes and no, it depends on your trip really. Can be helpful if your digging out a tent site, or if your in avalance areas. If your pack has an hdpe frame sheet they make good make shift shovels as do snow shoes. If your only doing a small area the improvised methods work fine, otherwise you need a real one.
Other gear that is quite useful for winter:
-white gas stove
-2 liter pot for melting snow
-matches, it can be challenging to use a lighter at times to light a stove etc. Matches work much better for winter
-dry gloves for camp(i like a fingerless for camp tasks, or one of those mitts/fingerless combos)
-A hardcore fire starting method for emergencies. I bring 2 sparklers(fireworks) cut into 3 in pieces. They have thermite and magnesium in them and burn about as hot as possible for a material(3000F) and last for a good minute or so and can light even wet tinder. And they are very light weight.
-if out for a longer trip I like to have a weather radio
If you have any other questions feel free to ask