sure, just get a little capfull to dab the toothbrush in.
the part you remove from the washer you can just rinse off.
you can wipe off the inside part of the washer when you are done.
if you want to go the pricey but easy route, they make HE washer cleaner tabs. I think tide, whirlpool and weveral others. I've never used this.ymmv
I think one of the most important things when washing is to make sure you are not overdoing the detergent. I remember a good rule of thumb is that if you see suds, you are using too much. use a little less next time.
We actually cheat here and use regular powder detergent. When we got ours a couple years ago, the tech who brought it recommended using 1/8 cup of regular detergent. Seems to work great for us.we have no problems with clumping, smells or gunk. When you use too much detergent it tends to stick in the washer and not rinse out fully.
I usually clean the dispenser drawer by soaking it in hot water. The hard part is getting deep into it. I use the sanitize option on our washing machine which helps alot. The water doesn't smell that way.