Not sure if this is a joke or not. MTB tyres are usually a soft compound, and cracking is normal after 10 years. Buy some new ones.
He dunalf talk some crap, this one.

SOME MTB tyres are of a 'softer' compound than others, but there is very little difference between bicycle tyres, as to the type of rubber. Kept inside and looked after, pumped up properly etc, bike tyres can last decades. It's leaving them exposed to big temp differences, damp, UV from sunlight, various chemicals, oils etc, that can cause degradation. I've got old tyres that must be approaching 20 years old, somewhere. Very knobbly 'mud' tyres of a slightly softer compound. Absolutely fine. Used loads of tyres that have been well over 10 years old, no problems. I've got my mum's ancient early 80s Triumph roadster in the basement; tyres are original and absolutely fine, cos the bike has been kept indoors all this time. Another thing that causes damage, is leaving tyres flat for ages; the weight of the bike sitting on the sidewalls can cause them to crack. Look for 'crazy paving' type cracks on the sides of the tyre, that's an indication they are going. Sometimes, the outer rubber starts to delaminate from the inner carcass, this can be potentially dangerous.
A pal of mine works in the cycling field building wheels - he was shocked by this too and recommended new ones
Lol! Not surprised! Big danger with those, would be possible delamination, and whilst the inner carcass should be strong enough to hold it all together (it's usually a woven material of some sort, nylon, kevlar etc), the tyre would be far more prone to being punctured, and would deflate a lot more rapidly if there's no outer rubber. Those tyres would also be deemed illegal to use on any public roads or cycle paths. I have seen worse, though, shockingly.
Retro bikes ??? That's a thing ??? lol
Yep. I have one MTB from 2000, one from around 2007, and another that is probably of a similar vintage. Got a 1990s road bike as well. Older bikes tend to be easier to fettle, and a bit simpler; current stuff is all fancy 11/12 speed gears, electronic systems, components that have proprietary fittings, and stuff that is really quite flimsy and wears out more quickly, (thus necessitating buying more stuff, kerching). Cost a bomb, too. Give me a 'retro' bike any day.
And here's the bike for you
That's alright, that. What I would suggest though, is replacing the brakes with a v-brake system. Easier to set up, and much more efficient. And make sure you check all cables and stuff for corrosion.
it doesn't like downshifting, so that takes a few attempts but usually gets there
Do you mean to an easier gear? IE, the chain moves UP the rear cassette/block? If so, you probs need to wind a bit more tension on/adjust the rear shift cable. Sounds like the whole system could probably do with as good service tbh.