I recall this subject being brought up on Y!A back in 2007-2008 when all of Edge's victories were either against Funaki or by cheating. Does it debase a World Champion to have to resort to these tactics, even if they are heel?

It depends on the character. If a monster heel has to cheat, that's bad. If it's a cowardly heel, it's gravy, as long as he's running away instead of doing the job on weekly shows (cougheveryheelICChampcough). It's just that, eventually, you have to give the guy at least one high-profile, decisive victory to legitimize the reign and make him a somewhat respectable athlete, or people will begin to question his worth:

I'm fine with it, so long as it's not the only way they win, and they don't do it too constantly.  The only heel who constantly won through cheating that I didn't have a problem with was Edge since that was his gimmick.

And I think monster heels cheating to is ok, so long as it's done properly.  Like if they're against another monster/big man it could work.  Also the manner of their cheating could play into their gimmick, like Bray Wyatt cheating by using mind games.

It's not cheating.

I think it's dependent on the performer and the character involved. Some people like Edge, William Regal, Eddie Guerrero and going back further like Roddy Piper, Flair, Abdullah the Butcher, they make it work because they add little tricks and tidbits to keep it fresh or interesting.

Then you have some guys who do thinks like HHH, JBL, Seth Rollins....where I just think it's a bit meh.

It's part of their character so it happens but really if it just keeps going on over and over again then it will get old and so will the character