Yeah, a little impactful, isn't it? (or, aren't they)? and as soon as you notice either issue bothering you, you start paying attention... and that makes it both issues so much worse.
There's only so much you can do without changing out the unit itself (and there are newer, quieter versions coming on line, including inverter ACs and heat pumps). As others have indicated, for your existing system, you can: (1) add a soft start - that will significantly reduce the initial inrush (and kill the big 'WOOMPH" of startup), but you'll still hear the system running; and (2) install the plenum channels (either the dedicated solutions or the do-it-yourself foam board built 'wedge') that the blocks off the center dump and better forces the air into the ducts to improve flow - and subsequently reduce a little noise. Both are better than not doing anything.
With a towable trailer, all this undesired motion is a function of being in a box on a suspension with mere stabilizing braces at the corners (not leveling jacks, mind you - those support the weight of the coach; stab jacks just work to
stabilize the coach, and let the weight be carried on the suspension and front jack). You've got jounce (sort of a combo of jolt and bounce) and shimmy (side to side rocking) - often all together, it's a confusing and somewhat upsetting feeling for your body to be moved in accordance with some one else's efforts. You won't notice it nearly as much if you're moving, too, and - even less so if you're the one causing these motions.
The good news is that everything you do to brace your trailer will help to some degree - though some things more than others. 1.
Chocking your trailer. 2a.
Putting down the stabilizing jacks (firmly, but not too tight - you can't lift the coach, just shore it up - these help reduce up and down motion, but not side-to-side) and - lesser known - 2b.
Limiting the height of the stab jacks (the FURTHER the stab jacks are deployed, the longer the lever arm is, and - subsequently - the less successful the jacks are at reducing side-to-side motions... not that they are very good at doing this in the first place). You're likely already doing these things.
Now, take it a step further: 3. I
nstalling X-chocks for multi-axle trailers to tie the axles and suspensions together (these are REALLY not as much help as you'd like, but better than not doing them). 4. S
tabilizing the stabilizing jacks (with crossbracing like the JT Strongarm, SteadyFast or even the Lock Arm system). Basically: firm diagonal bracing means fixing the stab jack foot in position and a LOT less slop in your stab jacks. A note on these systems - (A) the bigger and firmer the triangles formed, and

cool: the more dimensions the support is added in (fore and aft AND side to side), the more stable your RV will be. Systems that (C) more directly attach to the very feet of the stab jacks, (D) more positively the arms lock at desired length and (E) have the least amount of slop in the system will provide the greatest additional stability. Lastly, 5. Fixing the coach's frame to the ground. This is about as out of bounds as one can get
These steps are listed in order of ease and inverse cost - the further into the list you get, the cost and effort goes up, but so does the amount of stability you'll gain. IMHO, the stabilizing bar systems will get you as good a result as you can reasonably get.
There are other things one can do and other systems one can employ, but I think the listed steps above are a pretty solid approach (eg, one could also make a do-it-yourself bracing system with angled wooden crossbeams that are attached to the frame at two points and tied together with some strapping, but that's a different approach).
Just my .02, there's likely a lot of stuff that can be said on the topic. Hope this helps. could be wrong, but I believe what I am feeling is the floor itself flexing.