Should I cover area where ceiling once was?
There is no reason to be demoing this more than a day before it needs to be done.  Demoing the ceiling is a 1 hour job.  If they plan on running new electric and you will be demoing it is normal to cut out parts of the ceiling to make your runs - even big parts.  We would simply screw the drywall back in where we cut it out with 4-5 screws - doesn't need to be perfect.

Also you should cut out a 2'x2' hole, dump drywall and insulation in large trash bag.  Then get in attic and throw all of the insulation in the area in that hole - with a helper this is 20 mins.  This will give electrician room and view to do what they need to do.  Or you could just move insulation to other parts of attic.  But get it out of the way either way before starting to cut drywall as cleaning that up out of everywhere is a waste of time - and if your house is finished it gets everywhere.

So you can in effect have the bathroom cut into 4-5 small sections - effectively demoing it while leaving just the drywall up - just pop out the drywall over the screws using something flat and strong.  Yes it will have holes in it and not be optimal but you can still get it back up there.  You don't need to buy some special tarp for this.  Or you can just take it down an hour before they get there.

Notes:


There is nothing wrong with putting a TV antenna in attic like that.  If it works it works.  There is absolutely nothing ghetto about this.
From your picture there doesn't look to be any significant "shoddy wiring" going on.  If you are paying for "shoddy wiring" to be fixed I hope it is a real issue, not just electrician telling you it could be better.  But I have seen plenty of attics with "shoddy wiring", this looks pretty tame/clean.
There is no indication that the insulation in the picture contains asbestos.  It looks like your run of the mill fiberglass blown in insulation from the 60s/early 70s.
However while you are at it I would beef up your insulation.  Right now you probably are below minimum for your area.