Build Advice need help building small PC/server on miniITX mainboard

[McVitas]

Hello, so I have this little case LC-POWER LC-1370BII Mini-ITX with an old Core 2 quad Q8200 socket LGA 775 which I use as a home server mostly for file sharing, backup and music streaming.
I am thinking about upgrading it, so it would handle running a Minecraft server also, which I currently host on another machine which is Lenovo Thinkpad T420 with core i5 2450M. It works fine for a handful of players, but sometimes it already struggles.

This case is just 7cm high and there is only around 5cm clearance above the mainboard, so if I would build a new system in it it would need some low profile cooler (like AKASA AK-CC1101EP02). So I am thinking what kind of CPU and motherboard combination would

• work fine with such small cooler
• be relatively low power
• be faster then core i5 2450M which is on my old laptop
• CPU or Mobo should have integrated graphics because no card would fit (right?)

Maybe I will realize it's not worth trying to fit new hardware in an old small case and I will end up buying some new miniITX case, but I would rather not.
Maybe some Ryzen 3 or core i3?

 

[McVitas]

so it looks like Intel Core i3-10100 would be a good choice... but is there a miniITX mainboard for it??
EDIT: there is but it's 3x more expensive then mATX! So I might just buy a new case for mATX...

 

[McVitas]

well, it seems my topic is just too advanced for Tom's Hardware, so nobody dares to comment, not to say something stupid?...