4 GB Machine
At least for Alzheimer's, 4 GB of RAM, theoretically, is probably enough if folding is all its doing. My Alzheimer's CPU unit is using about 330 MB of RAM, and the GPU one varies a lot more but I've yet to see it exceed 1 GB. It may well be too little for some other types of folding though, so you might have to set preferences to avoid too much RAM usage.
The 3770K should be able to do sustained CPU folding so long as the mobo and cooling are in good enough shape. It should be able to fold until the cows come home, or at least until the CPU fan gives out. Of course neither one of our CPUs will do half as well as a modern CPU...
Big Fans Machine
About the extent of my ability to support Linux is "try it and see if it works". I've run into some issues with support for older graphics cards on Linux over the years, but that's more like 2021 Linux having issues with 2007 nVIDIA cards, or switching Radeon drivers resulting in the display no longer working. I guess I would say if the default drivers work, don't change them, if they have issues, try the proprietary drivers (especially if you have an nVIDIA card). Switching to the proprietary nVIDIA drivers took one of my mates' Linux Mint installs from literally unusable for gaming to not-quite-as-performant-as-Windows-but-satisfactory.
I'm guessing given the age it's probably on Windows 7 currently? I'm not sure that Microsoft ever actually closed the upgrade-Windows-7-and-8-to-10 pathway, initially it was going to be one year but I know they left it open after that. So if the concern is running an outdated version of Windows, it might be worth checking if you can just upgrade it to 10 and not have to worry about Linux compatibility with integrated Intel graphics from the early 2010s.
I see you're climbing the lists, just passed Louis Rogerson! You must have found the old passkey?
I've roughly tripled my point total this month, and have climbed from 17th to 14th. Still not getting any bonus points, but it's hardly noticeable relative to the points from 10 years ago. The RX 480 finishes one or two 70K point units per day, the 2500K throws in several thousand more, and the old Mac Mini gets 865 points about every day.
The real question is what is @Naskra
running to get those beastly point totals? A few beefy graphics cards? An underutilized server rack or two? I don't expect to hit those numbers even after rebuilding my rig later this fall, and even if I go towards the higher end of the specs I'm considering. Though admittedly I'm definitely not buying an RTX 4090.