well based on that video, i'd say you made the right call on the upgrades. CPU fan isnt working, so maybe a quick fix coulda been a new CPU fan to see if it'll cool the cpu to allow it to work if not fried. CPU's are really hard to kill, but it happens. Wouldn't make sense to buy a chip that fits that old motherboard, better off purchasing new ram, mobo, cpu.
Also that psu is an ultra LS 500w power supply, basically a no name chinese cheap junk (ShenZhen RuiSheng Yuan OEM), so yes i would replace it to any cheap seasonic bronze psu as it'll be a ton more reliable and safer. Especially if you plan on adding more components that adds more power to your system (ssd, new cpu, new gpu, etc.etc.)
http://www.realhardtechx.com/index_archivos/Page2713.htmi would not wanna put your new investments relying on an overrated 500w powersupply that barely pulls 336watts and isn't certified, but that's just me.
also u have a mATX board, but your case should be able to fit atx boards as well. Tho if you wanna upgrade to a new case with better airflow, this case is pretty good for the $$ atm.
https://www.amazon.com/dp/B075J6YSK2/?tag=pcpapi-20