iMacs run desktop CPUs, but mobile GPUs (which are still pretty fast, but not the fastest).

Apple REALLY needs to come out with a mini-tower Mac supporting desktop graphics cards. And no, I don't mean the new Mac Pro.

Yes but it becomes really noticeable (for me at least) when you get down the the final 5-10% of battery life.

A lot of laptops, by default, are set to save power significantly more aggressively when running off battery - it's most noticeable when attempting to watch netflix or play games on a cheap laptop.

Some laptops come with a powersupply that isn't good enough to run the CPU on full speed so it will even throttle down when plugged in... (HP I'm looking at you...)

If you have battery power left, it might run faster and drain the battery until it's low and then throttle down. Or just shut down like an older Samsung I saw once.

Yes.

Discovered this recently while working on a GPU-powered ray tracer, plugging my laptop in more than doubled the performance.

There's a separate line of processors for laptops typically, with Intel it's their U line, which have lower TDP and lower core count by default (i5-XXXXU have 2 cores 4 threads by default, a desktop i5-XXXX will have 4 cores 4 threads).

The prohibitive nature of battery powered devices has pushed Intel's processors to efficient low-powered solutions, but that's a big reason you'll see desktops simply overpowering laptops at the same price range.

Not to say you can never get strong processing in laptops though, they do offer the high-end HQ line which are high TDP mobile processors. Just that the default for Intel mobile processors is "gimped" by design to allow for efficient low-power processing.

Pretty much any laptop with a high-power processor (HQ series) will perform better when plugged in. You can force them to run at higher performance off the battery (go into Power Settings control panel, select "high performance" mode), but they will burn through a battery in like 20 minutes if you're really pushing them.

Sometimes Ultrabook-style laptops will have a higher boost, but they're often thermally limited anyway so running more power just makes them throttle harder.