How do I know if my solder joints are actually any good?

Yeah, that balling up is a heat thing. Solder flows toward heat, so if it's sitting on top like a little pebble, the wire itself wasn't hot enough to accept it. You probaly melted the solder against the iron tip instead of against the joint. The trick is to touch the iron to the wire for a second or two first, then feed the solder into the wire, not into the tip. When the wire is actualy hot enough, the solder will suck itself into the twist and spread out, and it'll go glossy. Dull and lumpy ussually means it froze before it fully wet the surfaces, which is a cold joint and will fail over time.

Gonna push back on the 'a second or two' timing there. Lamp cord strands are chunky, they soak up heat way slower than little electronics wires. On stuff that thick I count more like five or six seconds before the solder even starts to take, and I've had joints that needed longer still. If you pull at two seconds on big copper you get the exact pebble problem you were describing anyway. Feed it slow and let the wire tell you when it's ready, not the clock. Also on the headache thing, yeah that's the rosin, keep your face out of the plume and it stops happening.

On the holding question, a cheap helping hands thing is worth having but honestly half the time I just wrap the wire around a screwdriver or clamp it in a big binder clip taped to the bench, which costs nothing and grips better than those wobbly alligator clips ever did. The clips also conduct heat away into thier little arms, so a joint near a clip takes longer to get up to temperture, which matters when you are alredy counting seconds on thick copper like the second poster said. If you do get one, keep the clip well back from where you are actually working, and pre-tin the wire ends first so the final joint goes fast and the insulation never has time to cook back.

On the smoke thing, yeah that headache was probably the flux. Rosin fumes are a real irritant and breathing them up close is exactly how you get that foggy-headed feeling. I put a little desk fan behind my soldering spot so it pushes the smoke past my face instead of up my nose, costs nothing and works better than just cracking a window. If you do this a lot it's worth looking at one of those fume extractor gizmos eventually, but a fan does the job fine for occasional stuff. And don't lean in so close, your eyes will thank you too.