Verified answer

No, the preposition is necessary. "I need a sheet of paper to write" could mean you are about to start your novel. "I need a sheet of paper to write on" means you want to make a note or jot down a phone number.

"I need a guy to talk" means you want him to divulge information. Police want all their suspects to talk. "I need a guy to talk to" means you want a conversation. "I need to buy some toys to play" means that for a particular game or event you need to get specific toys in order to take part. "I need to buy some toys to play with" means you want to pass the time by playing with toys.

how are you

In some cases the meaning would be the same with and without the preposition, but that's because there aren't other possible meanings (for example the paper would be for writing on), not because the preposition can be omitted.

For your first and third examples, what you're omitting is "in order to". I need a piece of paper in order to write. (Otherwise I can't write.) So as I said that could mean the same thing, that you're asking for a piece of paper to write on, so that you can write, but it could mean something like you don't feel creative typing on the computer, you can only do creative writing by hand on paper (I guess this is what tentofield meant).

Same for "buy some toys to play with/buy some toys (in order to play)" yes the same event but the second suggests that you aren't able to play without toys, like you'd be just sitting there staring into space.

I need a guy to talk/talk to are usually completely different. "I need a guy to talk" would normally mean "What I need is for a specific guy to say some specific thing". It could possibly mean "I need a guy in order to talk" but that seems unlikely.