Yes, Of course the bus stops at Immigration but the point is you CAN buy a bus ticket online from Mukdahan to Savannakhet, Laos, thus providing onward travel by bus.
If you don't have any form of your own transport then you HAVE to take a bus across the Friendship bridge at Mukdahan. Walking across is forbidden by Thai and Laos Immigration.
Incorrect. There are buses that cross the border. Ubon to Pakse(Laos) via Chong Mek, Mukdahan to Savannakhet(Laos) via the Friendship bridge, Ubon to O'smach(Cambodia) via Chong Chom are 3 that I know of. I'm sure there must be more.