I have cold feet also due to neuropathy
Two questions there - yes to the cold feet.
Mine have been "really cold" inside or outside, summer or winter, for as long as I can remember and just about every doctor I have ever seen that checked my feet has asked "it's it normal that they are this cold".
I assume it is crappy circulation so as a diabetic now I am much more vigilant about checking my feet regularly and wearing protective shoes if there is even the slightest chance of bumping/whacking/dropping something on them.
Peripheral Neuropathy, particularly in the feet/legs effects 1 in 2 type 2's to some degree during the course of the disease and if you are like me and have "ice blocks for feet" which likely signals less then optimum circulation to start with, then neuropathy is probably even "more likely".
Yes to your second question as well - many diabetics experience some sleep disorder.
Having a sleep disorder or not regularly getting enough sleep (7 hours) in the first place puts you at greater risk to develop Type 2 (so there is already some link).
Then as a Diabetic you will see a higher incidence of Obstructive Sleep Apnea (stopping breathing periodically while asleep which makes you restless and can't get into deep (REM) sleep).
Some do try the (over the counter) supplement Melatonin which can help you get to sleep and stay asleep but you have to be careful with the stuff as a diabetic.
If there is still some quantity of it in your system when you wake up it will mess with your blood sugar. When your (body clock) biorhythm knows you are awake it doesn't want melatonin levels to be high - if they are it wrecks your insulin resistance and you blood sugar shoots up.
So best to ask your Doc "first" but if any try to self medicate, do read the bottle and take the stuff early enough in the evening that it will be all burned up "before" you wake up - you don't want to blow up your blood sugar trying to get a good nights sleep so that you don't blow up your blood sugars by being tired 😁