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You say you can get it for free but does that include the motherboard and DDR4 memory that you will need to go with it? If so then go for it just to have a newer platform with more feayures. If you have to pay for anything then it just won't be worth it and should wait until you can make a much bigger jump to something like a Ryzen 3000 series or at least Intel 10 series.
As long as the temps are ok I wouldn’t worry. It might be worth reposting the CPU and cooler though, any paste that was there is likely rock hard by now.
It's not an office system with a proprietary PSU is it?
If it's a HP 800 G2 or HP 600 G2 it will use a proprietary motherboard and PSU (as the motherboard has a smaller 12V only connector, and then rails for other devices e.g. Sata then "daisy chain" via connectors from the motherboard)
If it's a HP 400 G3 then it looks like it should be a standard ATX connector.
Regardless - it's still worth having for free - a 6700 chip on it's own is worth up to £100 still on ebay
Will depend entirely on the model, but if it's a 600 G2 or 800 G2 then no, as they do NOT use a standard 24 pin PSU connector (they use 12 volt only PSUs to comply with stricter efficiency regulations)
I'd either buy the latest thing you can comfortably afford, or not bother.
Alder Lake boards will support Raptor Lake CPUs, so I'd buy a cheap Alder Lake CPU if I was building right now.
I agree with some of the other comments, sell all the old kit and build a new system, upgrading to a slightly better system wouldn't be worth your time.
Personally, I would wait for end of 2022 for Zen 4 /AM5. This series will offer 8/16 cores, unlike Alder Lake and the Intel 13th generation (limited to 8 high performance cores plus additional lower power E-Cores).
But if not, cheap Alder Lake CPU (really depends how much you want to spend) + LGA1700 board, then upgrade to Raptor Lake later this year.
Intel's competitor to Zen 4 is Meteor Lake (14th generation), which is based on Intel's 7nm EUV process, which was delayed (originally 7nm CPUs were planned for 2022).
Have you confirmed if the system is non- standard?