I've got no idea if household soap will work or not, I just bought a bottle from Moleroda.

The way commercial systems work, is they periodically inject some water/soap in, and there's a drain plug at a suitable height to let excess run out.

I think the drain plug on the tumbler is at a suitable height for a similar setup, but you are correct in that the supplied tap is a rubbish fit. I bought new fittings to create a more robust setup, however it's still on my roundtuit list.

For filtering, letting the run off sit in a container would probably be enough to let the vast majority of debris settle out.