Je mange plein de barbe à papa

Maarten is right: use the plural if it's a countable noun and the singular if it's uncountable.

Je mange plein de chocolat. -- I eat lots of chocolate. (uncountable, since chocolate is a class of sweets)
J'ai mangé plein de pommes. -- I ate lots of apples. (countable)

As far as I have seen and learnt plural numbers will be used after beaucoup de/plein de, so "plein de chocolats"

It will depend on whether you are talking about countable or uncountable nouns, and for whether the noun is invariably or usually used in the singular or plural in the context. You have given some interesting examples and there are more !

Modifying answer but running out of time, so will get back later if this hasn't been addressed fully and correctly by anyone.