Hello, well for starters, pulling off 1 battery cable should not have killed the engine. The alternator should have kept it going, so in addition to the rough running condition you also have a charging system problem.
As to the rough running, I assume the engine rpm is fluctuating in tune with the Hz meter fluctuating? ( indicating a mechanical engine problem rather than an electrical issue with the Hz meter)
My first thought is a lack of sufficient fuel pressure or air in the fuel system.
Being the machine is new to you have you gone through the filters etc.?
I think you need to check the fuel supply and filters to make sure everything is ok there. Keep in mind that a fuel pump that makes noise doesn't mean it is working correctly.
If the pumps and filters are all ok its possible that just running the fuel pump in the prime / run position may clear it up, if you have air in the fuel system.
Likewise, running the machine a little may also clear it up if the problem is just air.
While doing this, it's always a good idea to add a little SeaFoam or ATF to you fuel to act as a cleaner / lubricant.
Once the fuel system is checked out, if the symptoms still remain we can move on to the next steps.
As to the possible alternator issue, first thing to do is check the fuse in the gray fuse holder located to the left of you AC selector switch, inside the control cubicle.