There isn't really any problem with swapping X and Y, but it will change how you do your programming. Standard convention is that as you are facing the machine, X moves left to right, Y moves towards you and away from you and Z moves up and down. But for most gantry type systems X is normally the long axis. (as it is on the diagram you showed) That does not seem to match your machine. It also depends on how you want to view parts on your machine and on your computer screen.
Yes, you can have problems if the acceleration is too high for what the motors can move your system. But with the motors removed, it won't really matter. So run the motors off the machine and check that they are all going in the proper directions and move by the proper amounts. Once that is confirmed then you can move on to acceleration tuning. If you jog at a slow speed or by a small amount, the acceleration won't matter, either.
Long arc, short arc, heliarc and in-the-dark!