Rough idle when hot

[Ravnica]

I got my alternator repaired recently and since then my cars been very rough on idle when hot. I've disconnected the alternator and the problem is the same and also I've changed the spark plugs. it's charging at 13.5-13.7 roughly with a full load. The battery is new, belt fitted on it today. Internal heater not working though I've replaced the thermostat (probably a clogged internal heater). Coil wires are all fine i've checked them all 3 times. The engine light is on and it gives multiple misfires on all cylinders. then then next day it gave individual misfires for each cylinder 1,3,2,4.

On cold it works perfectly. When hot and the engine light comes on it works normal for a while then it goes chugga again. sometimes when coming to a stop it cuts out (1 in 50 stops roughly) unless I go to stop when going under 2k RPM roughly. when it cuts out (stalls) my power steering goes and my breaks if I release them and press them again theyre much tougher until I turn it back on.

Car turns on fine, as normal as always whether hot or cold. Brought it to 3 different mechanics - noone knows what it is. Won't go to another mechanic until monday so I thought i'd ask and see if there's any recommendations.

please keep in mind I'm not a mechanic though I'm not unfamiliar with the general in's and out's.

I'm driving a 2001 Toyota Yaris with 170K miles.