Good start. Here's the things I noticed:

1. You don't have a GPU. The CPU you selected doesn't have an iGPU, so you'll need to have a dedicated GPU for the computer to work. Otherwise, you'll want a CPU with an iGPU. I believe you can get a 5600G for roughly the same price. Its CPU power will be slightly weaker than a 5600, but it'll come with an iGPU, and for $20 less cost. And it should be compatible with the rest of your build.
2. BIOS shouldn't be an issue. Older models of motherboards come pre-flashed with the compatible BIOS now. You only have to worry about flashing the BIOS if your particular motherboard was manufactured before the CPU came out (since the manufacturer obviously wouldn't have been able to flash the BIOS for a then-nonexistent CPU). The actual process of flashing a BIOS is really easy, but you'll need to install a compatible CPU first to flash the BIOS. Just follow the instructions on the motherboard manufacturer's website (or manual). It's pretty idiot-proofed
3. You'll want to buy speakers or headphones. Modern computers don't really need sound cards unless you're doing audio work, but you'll still need a way to hear the audio. (You'll also want a mouse, if you don't already have one)
4. 8 GB RAM is really not enough these days, even if you're using Linux. 16 GB is really the lowest acceptable amount nowadays. Doubly so because DDR4 RAM nowadays is really cheap. For just 20 dollars more, you should get 16 GB, DDR4-3200. It'll save you a ton of headache later.
5. You should get an SSD. SSD's are getting cheaper. Granted, hard drives are still cheaper, but it's getting to the point where the increased speed of an SSD easily offsets the increased cost of an SSD. SSD's significantly improve things like time it takes to boot up, time to open video or picture files, time to open programs, etc. It's really kind of a no-brainer nowadays, even if you don't game. I would not recommend a hard drive-only build nowadays, simply because the performance-per-cost is no longer viable. You should either do a full SSD build (more expensive) or an SSD+HDD combo (less expensive, but requires you to know where to put files). I would probably recommend getting a 1 TB SATA SSD (Samsung 870 EVO, $40) and a 1 TB HDD (WD Blue 3.5" 1 TB 7200 rpm, $30). This combo should get you the same total storage space, but it'll be much faster, and for just $10 more. Alternatively, you can consider if you even need 2 TB in total, and just get a 1 TB SSD. A solid 1 TB NVMe SSD goes for $60 nowadays.
6. Have you tried using that keyboard before? My personal experience with membrane keyboards has been generally negative. They feel really mushy and they cause a lot of typos because you need to press all the way down for an input to register. Perhaps it's not an issue at your current price tier, but I would recommend at least considering a mechanical keyboard. Despite all the stereotypes about mechanical keyboards being loud, there are many types that are designed to be quiet, and many more that are designed to do all sorts of things. I personally can't stand the clacking, so I use a mechanical keyboard with red switches (quiet). Keychron sells high-quality mechanical keyboards for relatively cheap. I would recommending giving their catalogue a browse.

If you're going for an iGPU like the 5600g as mentioned in another comment, you're going to want dual channel RAM. That'll add $11 to your total. The 5600g actually appears to be $9 cheaper than the 5600 so that's only a net increase of $2.

For about $10 more you can get a 1 TB NVMe drive instead of an old spinning drive. Kingston NV2, Solidigm P41 Plus, TeamGroup MP44L should all be fine options. It makes such a huge difference and is absolutely worth it.

Unless you're married to that size of monitor there's a fair few high refresh rate 1080p monitors with VRR support at or below the price of the one you have spec'd out. While you may not take full advantage with the build you have now, it gives the monitor the ability to grow a bit with upgrades you may do in the short to mid term. If you go with the cheapest one on that list, the MSI, that gives you $30 that I personally would put towards a more robust PSU, either this EVGA 600W 80+ Gold or this Silverstone 550W 80+ Gold.

As for your case, that one appears to only have 1 included rear exhaust fan, so you'd need to pick up some more like the Arctic P12 Max (9.99 single or 35.99 for a 5 pack). If you want to save a little you could consider the regular P12 (non Max, 8.99 single/34.97 5-pack) or the ThermalRight TL-C12C (6.59 single/11.90 3 pack). 2 intake fans with 1 exhaust would probably be sufficient. To offset the cost of added fans you could switch to this Cooler Master case.

Alternatively you could go with another case with more included fans, such as this Zalman case. Or if you're feeling adventurous enough to try RGB on your first build you could consider either these options from Deepcool and Sama.

Here's the full list with my recommended adjustments, only $7 more than yours but you gain:

• the iGPU of the 5600G
• another 8 GB of RAM in dual channel running at slightly higher frequency (remember to enable this in BIOS once you get everything set up)
• a 1 TB gen 4 NVMe SSD instead of a slower but more spacious hard drive (again absolutely worth the trade off)
• a case with a side panel (acrylic not glass but better imo than the cougar one) and 2 more fans
• an extra 50W in the PSU and bumped up to a gold rated unit
• a 24" 170Hz monitor with VRR

Mobo BIOS upgrades are usually really easy. You can typically connect an Ethernet cable and simply instruct the BIOS to update itself on many recent models and this has been my experience with that brand.

I haven't heard anyone needing a dedicated sound card unless you're specifically doing audio workstation type stuff.

I'm not qualified to answer your other questions IMHO.

You should consider more than 8GB of RAM. I think you'll have a better experience with 16GB on a new build because office apps can be surprisingly chunky. It's a cheap upgrade that I feel would pay dividends. Also, not having a pair of RAM sticks means your memory won't operate in dual channel mode, cutting your potential memory bandwidth in half.

1. The tricky thing about updating the motherboard BIOS to support a new CPU is that the computer has to be running, which means you have to have an older, supported CPU installed to do it. If this is your only Ryzen computer, you'll have to borrow an older Ryzen CPU from a friend or something to do the flashing.
2. You do not need a sound card these days unless you are doing something unusual (pro-audio, retro gaming while being a purist on getting the MIDI synthesis exactly right, etc.)
3. I don't know about a way to check ahead of time, but I'm 99.9% sure those parts will work in Linux. In fact, it wouldn't even cross my mind to worry about such a thing unless I was picking out something much more exotic.

By the way, ditch the hard drive for an SSD, even if you have to settle for less capacity to meet your budget. Spinning-disk drives murder performance these days.

I also concur with the other reply about the RAM.

I think you could probably shave a bit of cost off your monitor and keyboard to compensate (a 1080p ~20" monitor can be had for $100 or less, and I wouldn't pay more than $10-$15 for a membrane keyboard, wireless or not). Also, did you forget the mouse or do you already have one?

New updated link

Looks good now; no notes. 👍