If a homerun is hit out of the park, can the other team make an appeal that the runner missed a base?

No, once the ball has left the park it's a home run which is automaticly advanced to every base. If he misses the umpire will most likley tell him to go and run the bases again.

As long as the next batter has not been pitched to yet, any member of the fielding team can take the ball & tag the runner (no matter if he's on the field still, on the bench or in the clubhouse). If the umpires noticed that he missed a base, the call is "You're OUT!".

Yes the defending team can appeal before the first pitch is made to the following batter. And if the batter/runner did miss a base, the run he scored would not count. In your example, the umpire must have determined the batter/runner missed touching home plate.

If this occurs when there are less than 2 outs, all other runners who were on base score.

If this occurs when there are 2 out, and there are runners on base, and the batter misses first base, no runs would be counted.

If this occurs when there are 2 out, and there are runners on base, and the batter misses second base, any runners who began running from second or third base would be scored, provided they both crossed home plate before the batter runner past second base.

If this occurs when there are 2 out, and there are runners on base, and the batter misses third base, any runners on base would be scored, provided they both crossed home plate before the batter runner past third base.

If this occurs when there are 2 out and the batter misses home plate, and there are runners on base, all runs score except the batters.

It used to be that the batter need only touch first base, but the rule was changed to make him "touch 'em all." When Mark McGwire broke the record, he missed first base, and had to stop and come back to touch it.

If it is a situation where there is a home run, with men on base, and any of the other base runners become the winning run in the game, if the batter misses any base other than first, he would not be out, but would only get credit for the bases he touched.

in my experience,it's not so much an appeal, as it is to just go and tag the base that the runner supposedly missed, usually a good umpire will have noticed and call the runner out....

Yes. Not only that but they can make a case for a runner who passes another runner while running the bases. It happened once to Hank Aaron and Eddie Matthews.

NO THEY CANT