I can think of two possibilities:
1) Curled paper. Paper in a tray tends to curl a bit when it sits; larger paper sizes are more prone to this. While in the tray the edges curl down which, when the paper is fed, flips around and ends up with an upwards curl on the edges when it's under the print head. This makes the edges prone to come into physical contact with the head as the head moves over the page. The print head nozzles are really just a lot of tiny tubes - there are no valves, per se, to keep the ink in. Only the surface tension of the ink keeps it from coming out each nozzle. Which means if the nozzles come into physical contact with the paper, the paper itself will draw the ink out through capillary action.
2) Too much ink in the system in general. If you've done a lot of borderless printing or head cleanings, then the printer can build up too much ink. Head cleanings especially dump a lot of ink through, and if the waste ink pads are full then it just ends up collecting on the head itself and can end up in the parking cup (a little rubber pad with raised edges that the print head rests in, intended to keep the head from drying out when it's not in use). Once you get excess ink in the parking cup, it can smear over the surface of the print head and transfer to the paper.
The rainbow effect you describe sounds more like #1 to me. It makes sense that A4 from the back isn't affected because a) A4 is smaller and less prone to curl anyway, and b) feeding from the back makes a straight through path for the paper. In any case, though, stop cleaning the heads - only ever clean the heads if a nozzle check shows missing jets. Head cleaning dumps a lot of ink through and you don't want more ink right now.
For problem #1 you might try flipping your A3 paper over in the tray to try and reverse the direction of the curl when it feeds. This might make the print head less likely to touch the page at the edges and keep the paper from siphoning ink from the heads. Will this printer accept A3 from the back? Try that with a fresh sheet from the package to see if the problem shows up that way.
For problem #2 you might try running a page through to try and clean out the excess ink from the printer. I'd create a page with two hair-thin black lines running down the sides near the edges. Just something to make the print head to move the full range back and forth over the page as it goes through. If there is waste ink on the head, then this might help get it off. You can run the same piece of paper through over and over, feeding it from different directions and flipping it over.
