Things I've sworn I will always check in the future:
Check the loft for insulation (just moved somewhere that had not a SCRAP of loft insulation and didn't realise until we'd moved in)
Check inside all the cupboards for mould (kitchen, bathroom, bedrooms if there's inbuilt wardrobes)
If there's a catflap and garden, there is likely flea/s eggs about
Check the boiler and thermostat are working together
Check the shower and taps all work properly, especially important to check the water pressure if it's an electric shower
Check where the phone/master sockets are for your broadband to plug into
If it's a terrace, speak to both neighbours. Ideally you need two people to look round a terrace, so you can check how loud you can hear each other from different rooms. If you can hear each other making small noises in the house, you will probably be able to hear your neighbours making loud noises from theirs (although an autistic child that bangs their head against the adjoining bedroom wall at 3am isn't likely to be a common issue that everyone will encounter...)
Also you can check crime rates online with the postcode