Finding A Replacement Motherboard Compatible With An i7-2600 CPU?

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Cisco001 said:

Size of case and PSU generally do nothing with size of motherboard.
i.e. you can put m-itx or matx motherboard in mid ATX case. You can put SFX PSU in mid atx case.

Anyway what is your existing case and PSU?
Is your existing PSU good quality one?

Motherboard, what exactly is the currently problem?
As mentioned by others, have you change thermal paste over the many years you own the machine?

Case - what is your existing case?
Did you install case fans for cooling?

Oh, that's good to know! 
My existing case and PSU are from an old Dell Vostro 460 (I think most of the parts are from that.) It seems to be a standard microATX size, but I'm not sure it was ever intended to have a separate GPU inside - it doesn't seem great in terms of airflow. The PSU is 350W, and I don't think it's in very good condition.
I can only use half of the RAM slots or windows won't boot, and sometimes it won't boot anyway - googling the beep code it makes suggests memory or motherboard issues, but the RAM itself is fine.
Never changed the thermal paste since I got it in 2017 (I didn't really use it for first 2 years.) It was sold as 'refurbished', so I presumed they would've done that.
The case only has one additional fan for cooling - I don't think there'd be room to add more.