You don't have to specify Biz and you shouldn't get "No available flights" in most cases. You might have to break down the trip segment by segment to find availability or flights. So for NYC-CPT you're better off searching NYC-CDG and NYC-AMS and then AMS-CPT and CDG-CPT to figure it out, as an example. You should get back a grid with flight numbers and then availability. Try a simple flight first if you think you're not getting decent results, just to make sure it is actually searching correctly.

And, yes, you have to put in a return date, as you've discovered. It should default to a week after the outbound which is sufficient for your needs.

The other thing - increased availability for elites - only really matters for the "home carrier" searches. So as a CO elite I search nwa.com for partner flights but I also search co.com for seats on routes I know they operate since I get increased availability (and I take advantage of that benefit quite frequently). For you the only time that would potentially come into play is with FB and it probably won't actually happen.