We got our lb the dual sided mattress from mama & papas so one side is firmer (for babies) and the other softer (for toddlers). We flipped it to the softer side not long back and he has gone back to sleeping through. We put a pillow in his cot but we don’t lay him on it when we put him down but sometimes when we go through in the morning he is on it and seems to really love it - now they are toddlers you don’t have to keep the cot free of anything unless you want it, yes you can put the pillow under a sheet if you prefer or opt for a zipped pillow case - I wish they did coverless pillows like they duvets. We just have the silent night toddler pillow xx
We introduced one at 16 months and he sleeps a lot better and had no issues
We've just put one in with our 16 month old but that's more to elevate her due to a really bad cough - we've just put one of ours under her for now and as you said put it underneath the fitted sheet.