My dad had a tiny bathroom but still had grab bars installed for the toilet and shower. The grab bar was in front of toilet so could be used to pull up. If I remember correctly the county actually installed them free–cheaper to install bars than have to respond for a fall or help.
A small shower is probably better off with a shower stool which can be adjusted for height rather a built in seat. Plus you can buy them side rails if needed. We carpeted the bathroom floor totally (tiny bath) with bath rugs which I cut to fit to prevent him slipping (easy to pull up and wash). Warmer too.
If a toilet seat needs raising they make more than one size–my dad was tall and the toilet low so his was ordered had a 5 inch raise (much higher than what you normally find). If there is enough room between sink and toilet you can buy rails to put on toilet.
My dad always said he was born cold. Drove my sis and I nuts taking care of him because he always wanted the house at 80 degrees–and then he’d sleep with a blanket. I sympathize with you.
Between WalMart, Amazon and Goodwill we were able to outfit the house pretty well. It’ll be difficult to know exactly what might be needed until he gets there.
Stairs are a concern. I do know many people have installed a stair lift with good results.
How high is the bed? Sometimes a higher bed is easier to get in and out of but if he actually would need a step? Ugh.