If you have a septic tank, there should be a line in, a line out and a cleanout in the lid. You don't want to dump in that cleanout, instead you should have a tee in the line that goes in. I suspect that your "cleanout" is a Y on the pipe from the house to the septic tank and that would be fine for dumping as long as you maintain a capped Y for future clogging issues. For your dump line, 3" or 4" is fine, the most common size is 4".
In my area they are pushing for installation of a manhole in the tank lid so it can be properly inspected. Yes, pumping and inspection is now required.