Red alert! The breaker in position 6 is wrong!
Looking at your photos, somebody slapped a 30A breaker on your
LIV/BED circuit in position 6. I guarantee you that connects to a 14 or 12AWG wire, and the person "upped the breaker" because they are having trip problems. Somehow the house hasn't burned down yet, but it's a disaster in the making. Turn that breaker off NOW and leave it off until an electrician can fix all this.
Start with surveying which outlets are on which breakers, and label each of your outlets for circuit identity and ampacity (15A or 20A - it's on the breaker).
The breaker positions are numbered in the panel, but you may want to avoid using position numbers since you'll surely be moving them around. You could call them A, B, C, Thor, Daisy, Yondu, etc. Whatever.
A circuit can handle:
A 120V/15A circuit can handle 1440 watts continuous, and 1800W total for short time (<3 hours with a long cooldown period between peak runs).
A 120V/20A circuit can handle 1920 watts continuous, and 2400W for short time.
You cannot use circuits larger than 20A.
If you have 10 sockets on a 20A circuit, that's like having 10 checks and $1920 in your account. If you spend your first two checks on $1200 servers, you're overdrawn even though you have 8 more checks. You know this.
240V circuits can carry double the above. We'll come back to that, but for now, 240/15A = 2880 continuous and 3600W max. 240/20A is 3840 continuous and 4800 max.
First, I know that EVGA power supply is in vogue. But we deal with it from a power perspective, and it's super impractical. You don't need 850 frickin' watts unless you're rendering or mining bitcoin with dual 1080's and a stack of server grade drives. Seriously. But the power-side draw of the pig is 10A requiring you provision 1200 watts (inefficient!) Dump that pig unless you have a critical-path business need for it.
Now survey each piece of your equipment, it will have a nameplate rating in either VA, watts or amps - VA is the best number to use, otherwise use the larger of either input watts, or amps x 120. Like the "850 watt" PS is 10A therefore 1200W.
Don't forget your other non-computing loads - survey the whole house, it's common for multiple bedrooms to share a circuit.
You know what to do now. Spreadsheet it all out, and figure how to dispatch the loads you have to the circuits you have. Obviously you will have a shortfall, but now you know how much.
Don't forget air conditioning. Every watt you consume becomes that many watts of heat in the building, which you will need to air-condition to remove, and it costs 1 watt of A/C to remove 2-3 watts of heat.
Thinking about 240V circuits
Your idea of splitting 240V into two 120V half-circuits is called an MWBC and it's too difficult to do given current codes. But it's just as easy to wire two full circuits. (you couldn't have used existing /2 cable for an MWBC, so you'd have to pull a /3; well then pull a /2/2 (4+ground) or dual /2's and done.
You can also bring real 240V into a bedroom, and that can run on /2 cable. However there must still be 120V outlets in all the usual places (within 6' of any point along a wall), so you can't re-task a 120V line unless it's an extra that isn't needed to meet the minimum service.
A 240V-only line can be run with /2 cable, but if you're in the walls you might as well pull /2/2. Most of your equipment will already work directly on 240V. Read the spec, most has multi-voltage power supplies.
All of this involves pulling new wire through the walls.
You never, ever connect a 240V circuit to a common standard NEMA 5 receptacle. They make NEMA 6 receptacles specifically for 240V (without neutral). They also make NEMA 14 for hot-hot-neutral 4-wire connections, but those do not exist in 15 or 20A, so they are irrelevant to you.
Upping the panel
Changing a main panel is a huge and very expensive project for an electrician. Avoid if feasible, especially since GE Qline is a fine panel. You do have far too few spaces, however, and the way to fix that is a subpanel. And please, get lots and lots and lots of spaces - 40 is not excessive.
Then you put a decent size breaker in the main panel (e.g. 70A) to power the subpanel. The subpanel doesn't need its own "main" breaker, but if it has one, it just needs to be same or bigger. (e.g. 40-space panels tend to have 225A mains, fine.)
If you have no empty space in the existing main panel for a fat 2-pole breaker, pick some small circuits whose wire length is favorable to moving them to the new subpanel; do so; and use the freed space for the new main breaker.
I would assume to put the new main panel right next to the existing main panel. Depending on what's possible with moving otehr circuits over, right in the server room may be a good place.
Load planning for the entire house
Anytime you make changes like this, you should also survey all the loads for the entire house and make sure that adding circuits is reasonable and legal. There are plenty of Q&A here and articles elsewhere on that.
The presence of a subpanel has nothing to do with it. The main and subpanel loads are counted together to determine if the main panel has the capacity for what you need. It's much more involved than adding up all the numbers on the breakers.
Separate to that, you also do a load survey on just the subpanel to make sure it isn't being overloaded.
Wait. You are rendering or mining bitcoin!??
Hoo boy. Start by seeking PC power supplies that are more than 70% efficient. 1200W for 850W is terrible.
Do your load planning based on your actual loads, and then when you wire it, install 2 circuits where you would've installed one - running 2 cables instead of 1 costs next to nothing, and it's in the walls for future use. But that's just more of what I talk about above.
Fair chance you'll need more service, and have that conversation with the power company. Again you don't want to replace that main panel. So fit your LARGE subpanel where new service can easily reach it - and at that point, flip it so the new panel is the main and the old panel is the 'sub. That'll work just fine.
Air conditioning is HUGE. Every watt turns into heat, in the room and you must remove it or the room will overheat and the CPUs can't cool. A/C is not optional here. Your whole-house A/C is not designed to remove that much heat from a single room. I have told bitcoiners who had an empty house to scatter servers all over the house, for that reason (also power). However there's a better way.
Use water cooling, and ship the hot water outside. These guys here did exactly that, because they had the same problem: five max'd out rigs in one room and the house A/C couldn't deal. It was very effective. But they made blunders:
- They did not insulation-wrap their supply and return pipes.
- Being a "review site", they are obliged to use the free stuff manufacturers send them, that's why they spooged together their radiator. They should've gone to an auto junkyard and bought an SUV radiator, with electric cooling fans, thermostat and relay. Which is way more than they'll ever need, and is made to go outside.
- They mounted the radiator in direct sun.
(and by "water" I mean 50/50 propylene glycol, a more rarely-made automotive antifreeze. You will have leaks, and propylene glycol, being basically a foodstuff, is nontoxic and no worse than a Mountain Dew spill. Common automotive ethylene glycol antifreeze is the wrong stuff, it's used because its cheeaper, but it is both toxic and sweet-tasting, so it will kill your pets.