While in Vienna we are planning on doing the day trip to Melk, returning via the river. Is it possible to rent bikes and ride for a portion of this trip?
Yes, it is possible. Look at nextbike (www.nextbike.at). Take the train to Melk; take the WL1 bus to Spitz; rent bikes in Spitz and cycle to Durnstein; get back on the WL1 bus to Krems; take train from Krems to Vienna.
If you have Rick's Vienna/Salzburg/Tirol book, then see p. 233 for a short discussion of biking either side of the river. He says it's a "3 - 4 hour, gently downhill pedal from Melk to Krems." Note that you need a phone or internet connection to pick up the Nextbikes.
Thanks for your input, I like the nextbike idea!