The problem I am having is the pipes that run along the ceiling sweat really bad and it causes me to not be able to put electronics in certain areas or anything that I don't want water damage on. Does anybody have any suggestions to stop the pipes from sweating?

The pipes are not sweating. The humidity in any room will always condense on the coldest thing in that room. Wrap them with the insulation. Waterproofing the walls will also help reduce humidity. www.ugl.com/drylok/...

We are thinking about moving the washer and dryer in our basement to a more logical spot. This would mean running our pex piping = not a big deal. My concern is more for the dryer - how long can the dryer vent hose be? Ours would likely need to be about 16 feet long to go up the ceiling and back out to the hole that's already there venting outside.
Thanks, George. That's what I figured. Much appreciated.

If you use solid vent pipe (not the flex stuff) you will be ok.
Just clean it out once a year.

I've been wanting to get going on this project for 3 years, but I'm just not sure where to start. I just want to transform my basement into a clean place for my kids to play, a bright place to do laundry, and some storage space.

Call some contractors and get estimates.

If you plan to do it yourself, the basic steps are:

Address any water/leakage issues
Frame out the walls
Install electrical
Sheetrock and texture
Paint
Flooring
Finishing.

My girlfriend and I have a partially finished basement. She wants me to divide part of the basement off for an office. What is the best wall/divider/partition I can use so she will have privacy from the rest of the basement? Is Drywall the only answer? What can I use for a nice ceiling as well with lighting? Thanks for your help, it really means a lot to us.

There are many ideas for dividing rooms. A wall is the best, but if you do not want that, you can use purchased office dividers (can be expensive), or bookshelves, or changing screen, or even a ceiling to floor curtain on a cable system. I like drywalled ceilings, but you can use a drop ceiling, or tiles. As for lighting..most basements have little natural light, so great lighting is necessary. Recessed lighting, track lighting both work well. If you do not like these then traditional lighting is suffice when portable lighting is added...table lamps, floor lights..even rope lighting in niches or shelving all add to the ambiance if that is the look you are going for. Brighter lighting is a must in the office section.