The pull through method is good. There are the long shove-it pads that stay in the horn but they tend to allow the moisture to dry up inside which can still cause problems.

Personally I don’t use anything except for running something through the neck to avoid any build-up inside.

The mouthpiece discussion has been mentioned here before but in short, the best practise is to take apart, clean and store the reed. Personally I rarely ever do any of that, just leave it on for next time. As a beginner it’s best not to do what I do because your chances of playing with a warped reed are high and you may have more problems because it it.

Stick to taking the reed off, wipe it and lay it tightly with an elastic on a flat hard surface such as plastic or glass. This will give your reed a longer life as well.