TTP44
My comments.
Is there a problem using road bike type wheels on a
recumbent? --Not really, however, you'll have to change the length of
your chainstays. Plans are for 24" mountain bike and 700c is about
27".
Are there real advantages to the fatter tires--Maybe on wet roads
larger tire contact area with the road. Tire pressure can be lower
so better road shock isolation. However, I ride at 100 PSI so more
input to the seat. My wife wants me to build one with suspension.

How many speeds are common for the rear? -- I have 11-34 rear
cassette and triple up front (52-42-32 I think) So I have Road
gearing up front and mountain gears in the back (all Shimano. Use a
large gear in the back you'll like in on long climbs.

What type of shifters work best?-- Could by personal preference
here. I use SRAM twist shifts I love them. You don't have to remove
your hands. All bents I have looked at use mountain bike type
shifters. Not sure how you would go about using road bike shifter.

I worry about getting my back and butt drenched with cold, dirty,
water or slush from the road.--Add a fender. The riding I have done
it hasn't been a problem. However, if you're in a all day rain it
could be.
Bob in AZ