General

Well the problem here is that its a board made for HP, they are the ones who may or may not provide BIOS updates for your motherboard to allow a newer/different CPU. So I think it would be wise to play it safe and stay with the compatibility list. As others have said the 370k would be a slight upgrade anyway, but probably nothing major, maybe if you jumped to a x4 740,750k, or 760k you would notice improvement it programs/games which take advantage of multiple cores. I wouldn't try overclocking on that board though, im not sure how much it can handle.

It looks like if you board says JasmineR-Version 1.0 on it there is a bios update that would allow the athlons to work properly.
here is a link to the board [url=http://h10025.www1.hp.com/ewfrf/wc/document?docname=c03757575&tmp_task=prodinfoCategory&cc=us&dlc=en&lc=en&os=4158&product=5399835&sw_lang=#N133]link[/url]

Yeah, you shouldnt have a problem with it not being compatible...It has the same wattage, same socket, should work fine.

But as has been mentioned, it isnt a very large upgrade...According to benchmarks, you will see about a 15% performance increase for CPU.
However, go up another price bracket up to the 100-120$ range, you can get atleast twice the power of your current CPU while maintaining the same wattage and socket.

As far as the listed incompatibility, it is unlikely that the one you selected would have an issue, but if you want to play it safe (Not a bad idea in any case.) then pick one from that list.

You'd probably would want to stick to the a-series processors as since that is what yours came with. If you want to upgrade, grab an a10. The website you linked to has a list of compatible processors. Pick one that fits your needs and buy it.

i think the cpu you're trying to replace your original one with is actually older. Should be the same socket, but im gonna guess that there's going to be 0 performance change, bc theyre about the same tier