They usually start taking lights and decorations down right after New Year's, but it takes quite a few days, so on Jan 3rd I'm sure there'll still be quite a few decorations still up. However, they may no longer light some things.
On the other hand, they may actually wait until Jan 5th and leave a lot going for the weekend and then actively remove them that following week.
The only thing I've ever seen published is the Osborne display dates. When they start coming down they close the whole area. There's usually signs when you first enter WDW saying what the Osborne display dates will be, or MS can probably tell you when you check in.
Weather in January can and does vary considerably, so no definitive answer can be given. You may find it in the 80's all day long, or you may find it never gets above 40 the whole time you're there. It's just like weather up North. One winter January temperatures are in the high 20's or low 30's, and the next winter you're looking at 10 below zero. Basically the warmer it is up north, the warmer it will be in Florida, and the colder it is up north, the better the chance Florida will be cold (relatively speaking....).
It will definitely be cooler at night so take a jacket with you even if it's warm during the day. Just hope you don't get a lot of rain as that can really make it feel cold.
Have a great trip.