Those corks should be good. They will mushroom...all corks that look mushroomed were original straight-sided.
I've never used a floor corker to cork belgians, so I can't really comment. If it were me, I'd probably pick up the much cheaper Portuguese Floor Corker ($60 usually I think) than the Italian Floor Corker. I have the Colonna Capper/Corker and can't really recommend it. The capping bell dents all the crowns (so I won't use it as a capper) and the corking is pretty annoying (the bottle jams and you really have to wiggle it to get it out...or insert a second cork to pop the bottle off).