The gear head seems to get very hot after only 5/10 minutes use - is this normal?
I get the same, its more so on a hot day, and it seems normal caused by the friction of the gears, but its within the design tolerances. You don't need to lubricate the bevel head, its filled with the right amount of grease at the factory.
What speed should I run at with the mulching blade, should I be going "flat out" the whole time?
Like a chainsaw, its designed to run at full throttle (see page 20 of manual). If going at slower speeds you'll have bigger bits of chopped material and more strain on the machine. Don't run it at full throttle without cutting or it'll spin very fast and overheat. The technique is idle, go to bit to cut and give it full throttle, back to idle when done that bit, move on, next bit at full throttle, etc.
Is it normal for the machine to vibrate more at slower speeds (mulching head again)?
Don't run it at low speeds. They do vibrate, particularly as they are hitting stuff, which is why it has built in dampeners. After an hour or so you'll feel some tingling effects of the vibration in your fingers, so don't overdo it. Just double check you've put the blade on right, it should be more of a high 'buzz' vibration than a low 'wobble' or 'rattle'.
I suspect you've not read the operating instructions, it always helps !
http://cdn.husqvarna.com/ddoc/HUSO/HUSO2008_EUenAPen/HUSO2008_EUenAPen__1151109-26.pdf