Set your layout scrolling properties to unbounded, and it will get rid of the snapping. Then use invisible walls or something to prevent leaving the layout/seeing black edges.
Sadly, setting the scroll speed % is probably the best way I've found of handling parallax. It does mean you'll need trial and error to make your maps, or create an in-game map creator though.
As for the bonus question, maybe have private variables for each object storing their start/old X and Y. When sprinting, move them to -10000,-10000 or whatever and then place them back at their starting/old X and Y when done. Have to make sure the player doesn't stop sprinting in the middle of one though I guess.
As for the image sizes, can you break them into smaller tiles? Like 512 by 256 or 512? It does mean lower quality images if you choose to just shrink the graphics and scale them up in-game though.
These are fairly short answers, so any further explaining I'd be happy to give when I next have time. Hope this helps though.