Since there is no dishwasher and it is a compact kitchen I would have a drying rack for dishes added over the sink.

Also I’m fighting a nasty cockroach problem thanks to tenants leaving food sit out and piles of dishes lay around .

Buying a Hulu/Netflix just for that rental would be cheaper than DISH but when I'm traveling I like to have access to live stuff.It doesn't sound like I'm missing anything.

I mean a true Petri dish of a house.

The real issue could be anything from its truly sporadic, dishes could be overloaded, baking sheets covering the soap dispenser, or the tenants expectations are just unreasonable.So at what point to do just replace it or tell the tenant that they'll be charged if you have another visit that finds nothing? 

Not the cheap ceramic soap dish, an acutal recessed soap area in the wall.

For example I have only college aged people living here, it'd be pretty weird if I had 20 year old girls and a 40 year old male.An important rule is establishing quiet hours, this helps everyone out in the end.Some of my tenants have complained about dishes and what not, but I encourage them to address that with their roomate, and it generally works out.

She has two dogs, so there are water and food dishes out.

Your tenant needs to not leave any food out on tables, counters, etc and clean up the dirty dishes. 

Take a small bowl, put apple cider vinegar and a squeeze of dish soap in it.