0.2mm DOC is disappointing. 12165rpm sounds a bit fast for an 8mm carbide cutter in aluminium. I use around 9800rpm (600mm/min, 1.5 DOC,8mm single flute). Looks like that was a 3 flute cutter since you've set the plunge to a third of feedrate which is what Gwizard suggests? If so then 750mm/min is slow, you wont be getting a big enough chipload. Use more like 1800mm/min if it's 3 flute.

If you want a quick way of guessing the feeds and speeds then scale it proportionately off one you know works. For instance I know that 9800rpm, 600mm/min, 1.5 DOC works well with the 8mm single flute, so if I used a 2 flute cutter I would just use twice the feedrate (1200mm/min), as its got twice the number of flutes and maybe decrease the depth of cut to reduce the force on the cutter. Using that logic, your 4mm 3fl should be run at 19600rpm and 1800mm/min...clearly this isn't the most accurate way to determine the settings, but it's a good start.

Sounds like you're describing and oldham coupling. They're pretty good, but you should be able to mount the motor accurately enough to use a solid coupling.

You could try using a finishing pass to eliminate the chatter marks - just take off 0.1-0.2mm at full depth.