Learn English – “Seeking for an answer” or “seeking an answer”

To me these two statements have different meanings.

What are the alternatives to this project?

This means "What other projects could we do instead?"

What are the alternatives for this project?

This means "In the context of this project, what choices do we have about how to solve it?

Engage with somebody means, as others have said, to interact with that person, usually from a position of greater power (managers are frequently exhorted to engage with employees, but rarely the other way round). Engage somebody has many possible meanings, depending on context: the army engage the enemy, you may engage somebody in conversation by simply going up and speaking to him, a pretty girl may engage (or behave engagingly to) the man she is interested in. The basic meaning is 'get involved with', which is similar to but not the same as the buzzword engage with.

And, as Barrie says, the usual meaning without explanation of engage a man is to hire him (at least in Britain).