Any Benefit to Short Bike Rides?
Not likely. It isnt long enough to offer much aerobic benefit. A 10 to 20 minute run doesnt offer much aerobic benefit, either, but still offers some CNS training and some strengthening of support muscles in a way that is running specific. So a little bit of biking isnt maybe a complete waste of time, but it probably is only a little better than nothing.
If you have weight issues (and most runners dont), this could be beneficial purely from that aspect.
just get a bike that fits you. No benefit to a short bike.
Every time I get to ride a bike I become happier!
The training benefit to riding a bike as you described it would probably be mostly along the lines of making sure you have a nice, easy WU or CD, stretching/mobility, etc. Auxiliary stuff to actual training, but important