Not sure if you want to hear this but this is better handled before finishing as supposed to mitigating after the fact. Either way I think you need to sand the roughness out you have first then do something to prepare the edges for finishing. In your case finishing would be painting.
Preparing Plywood Edges
Plywood surfaces are usually finish ready. That is not the case with edges. For larger project you can use banding/veneer or solid wood edging. In your case those would not be ideal since you are working 3mm board. What I think you should do is apply some compound along the edges. You have choices here but you can use wood filler/putty or something simple like drywall mud/compound. Your workflow should be something like this
- Sand edges.
- Liberally apply compound of choice. This will ensure you get gaps between the plies if they exist.
- Sand again to get smooth finish.
I spoke in the comments about the raised grain. This is the wood swelling from moisture contact. This is a normal occurrence and can similarly happen in engineered wood products. This is more dealt with solid wood products. Plywood still contains solid wood though. Read some of those links to Woodworking.SE that cover those topics in more detail.
When you raise the grain and go to sand it again you are just trying to remove the raised grain and not sand the surface of the wood. Doing so will expose more grain and you likely will have the same problem.
I am not convinced that is your issue from the picture. The edges of your wood there really look like tear out or blow out from the saw. Even if I am wrong the following suggestions would still be useful for the future.
When cutting wood like this you can try to prevent tearing of the wood fibers, on thin wood like this, by having tape on both sides. Something simple like masking or painters tape can help stop that. You still have to sand of course afterwards.
One other thing to keep in mind is that the family of saws used for this work (coping, fret and others) almost exclusively only work on the forward or back stroke. It depends how you have the blade mounted of course. Don't but force when the teeth are not going to cut.
Unfortunately I don't have any suggestions about the paint since I usually use acrylic for applications like this. However you could try thinning the paint with water if you want a more opaque effect.