Award Booking Help - One Way C3-C2-C1

Hello fellow FT'ers

This will be my first Award Booking and trying to get a little assistance. I realize that I need to do the leg work and make the calls, just trying to figure out what I should know before I make the calls to United (or whoever is best).

Im trying to go from HKT to BOI but route through Europe with a 6 - 10 day layover. First is this even possible using one way award ticket?

I will be flying with my 4y/o daughter and hoping to book C Class travel at the 60K point saver level.

So the questions:

Is this possible, or if not what is the longest possible layover I could have at a major European hub?

How do I search for open travel seats? I used United.com but I don't know how to search via the Atlantic route. I heard about an ANA tool, but can someone just give me a rundown or a link to a thread I could read on this topic?

What airline should I book the travel though?

When I ultimately have to call them, what things do I need to have ready so I can direct the call.

Thanks in advance to any helpers!!!