What kind of machine is it?
I agree a teacup should not require support, unless the handle has a loop that dips below the attachment point. But even then only the underside of that loop would need support.
The layering can leave stripes, but a nice material with good print settings on a well made and tuned machine it's more of a texture than actual visual artifacts. They're like grooves on a record; you can feel the individual grooves but unless you look closely or get the light at just the right angle it just appears as a matte finish.
I've only printed an object intended to be liquid tight once, and it worked fine. Again, it comes down to print settings, calibration and good quality material.
So in the interest of improving your 3D printing experience, I'd like to know what machine you have, what material you use and what the settings are.
As for speed, that's also generally a limit of the material... but I've gotten mine up to ~230mm/sec before the heater in the nozzle couldn't provide enough power to melt the filament at that rate.
In practice you have a lot of moving mass which limits your top speeds on complex parts - the forces from accelerations can overwhelm the cheap belt drive systems most hobby-level printers use. Of course, if you want to shell out for better parts you can make something a lot better :)
=Smidge=