Acid makes the OH a better leaving group, since the new leaving group will be the weaker base H2O, not HO(-).
Base makes the OH a better nucleophile, since RO(-) is a better nucleophile than the neutral alcohol ROH.

Both substitution and elimination reactions of alcohols can be catalyzed by acid.

This would be an example of anchimeric assistance (neighboring group participation). After protonation of OH, the phenyl group acts as an internal nucleophile, leading to a bridged intermediate. Attack of water on the bridged intermediate gives 2-methyl-1-phenyl-2-ol, which then undergoes a normal dehydration to give 2-methyl-1-phenyl-1-propene

Because in order for elimination to occur, the C-H bond has to break on the carbon next to the carbon bearing the leaving group. The leaving group is on C1, the CH bond must therefore break on C2, and the bond forms between C1 and C2, giving 1-butene.

Not conventional E2 reactions. Deprotonation of the hydroxyl group would make the resulting species (O-) an even worse leaving group!

It’s a way of forming a cyclic acetonide from a diol. https://en.wikipedia.org/wiki/Acetonide

Not in one step. It’s necessary to do a reduction of some kind.
However, there is a reaction called the Corey-Winter reaction that will reduce diols to alkenes. https://en.wikipedia.org/wiki/Corey%E2%80%93Winter_olefin_synthesis