- Can I just switch out the unmarked one for any one of the confirmed working ones to test if it's broken? (I.e. With a swap my outlets start working again)
You could if you used one of similar rating, but you need to be very safe (turn off all the house power ahead if time) and I don't recommend it.
- Does the gauge of wore going into each of the fuses mean anything? (I find it odd that kitchen outlet fuse on the lower right has a far larger gauge of wire than the two kitchen outlet fuses above)
The gauge of wire matches the rating on the breaker. The lower kitchen fuse might have some heavy appliances on it like the garbage disposal.
- What do the numbers on the fuse switches mean? (Most of them have 15, joined ones have 30, some have 40)
The numbers 15,30,40 are the current ratings of the breakers.
Since you have identified the breaker in question, open it (move it to the right), then close it again. If the circuit is still off, the most likely explanation is that the breaker is bad. You can find a procedure to replace a breaker on Youtube. It is not hard, but you NEED TO MAKE ABSOLUTELY SURE THE HOUSE POWER IS OFF. This means throwing the house breaker and using a volt meter to check that the power is off.