Congratulations on your little one! We have a 22 month old boy and well, he gets hot just like his parents.
I've always been told that a baby will let you know if they are uncomfortable due to external and internal temperatures. With that said, I do keep close eyes on his flushed cheeks during activity and offer him water frequently. I also recognize the change in weather temps and well, I'm sure you do this too...adjust clothing and blankets accordingly. I do not use the footie pajamas due to our son getting too warm; however, I do put socks on him especially during the cooler months. His blanket flops off him thru out the night, but I have yet to check on him and note that he is too cold or too hot. He continues to sleep just fine.
I realize our boys are distant in age; however, some children just seem to sweat more than others. And I would imagine your 3 month old is still regulating his normal body temp outside of your womb.
Our son also gets the wet head when he sleeps in his crib, but I think sometimes it's a combination of teething drool and yes, bed sweats (I still get bed sweats at 36!). I do use a small fan if it's too warm just to help the air circulate. We also use a humidifier especially when the heater is on. I don't know about the cooling pad, this seems a bit much to me and well, could pose some additional problems (too cold at the temples and ears). I suggest trial and error and give him time to adjust to his crib and new room. Observe and adjust, that's my motto!
If you are truly concerned, discuss further with your pediatrician. They may be able to provide some additional insight. Good luck!