First, welcome to the boards!
What an exciting time, new baby, first WDW trip!!
We have gone in Nov. a bunch of times and with our kids. DS was 9 months on our last trip in Nov. I do want to say that week after Thanksgiving might not be water park weather. We usually are there week before Thanksgiving and it is usually in the low 80's high 70's. By the time you arrive the temp's would be more in the high 70's. Might be too cold for the baby, eveyone is different, we like to swim only when it's hot, the kids will swim no matter what!
The baby care centers are so great! They have breastfeeding rooms, family bathrooms, any and all baby items you could possibly need. Changing tables, high chairs, t.v. with little tables/chairs and toys, coloring sheets, etc. It's a great place to unwind and refresh!
At 1yr. they usually are not scared of the characters. Of course some kids are scared but in my experience, it wasn't until 2.5-3yrs. the fear of the costume characters set in!
I think it's a great idea to bring your own stroller, we always do. There are umbrella strollers that recline. We bought the Kolcraft one. It is great and was only $30, it also has a sun shade and mesh bag for diapers and such. No hanging head during naps!!
As far as resort ASMo is great. As long as you don't mind bus trasportation to the parks. The resort is great for families and the price can't be beat for staying on property. However, we are spoiled and have only stayed in a Value resort once, for 1 night!LOL We like deluxe resorts for the bigger rooms and monorail or ferry access to the parks. Now we are DVC members, so it's home away from home resorts for us!!!
I think you will be fine and miss the Thanksgiving crowds.
I didn't breastfeed so can't help you on good spots to do so. I have seen so many moms breastfeeding everywhere at WDW! Most popular seems to be on dark rides!!
Your 1yr old will be able to go on so many rides and attractions with you. You will have a great time.
Be sure not to miss Playhouse Disney at MGM.
A helpful tip that helped me was taking my child to a local park to do a "test run"!!! It got them use to seeing characters and being on rides and crowds. Both my kids have been going to WDW since 9mo old! We go every year and we would take them to Hershey Park in the summer before our fall trip just as a test to see how they are handling things. It has really helped them have no fear of rides, crowds, even swimming!
Have fun planning and just read everything here on all the boards! By the time your trip comes, you'll be an expert!!