Assuming I satisfy all the visa requirements it is possible to cross from Nong Khai to Laos by bicycle? I want to cross back into Thailand departing Thakhek into Nakhon Phanom

They will not allow you to walk across the bridge. Not sure about a bicycle. If not perhaps you could take it on the shuttle bus with you.

I suspect using the (once daily) train might work.

There is always a local with a pick up. I am sure for a 100 baht he'd love to take you across the bridge.

It is no problem cycling across the bridge at Nong Khai, I've done it many times. The bridge at Thakek Nakhon Phanom however will make you put the bike on a shuttle bus.