You will usually clean it once a month is sufficient. If you notice slow draining or the washer gets stuck on a cycle many times that is indicative of the filter needing cleaning. Great idea to be pro active and glad you asked as many people fail to even realize there is a filter and call service unnecessarily.
So you will need to buy a front loader as they are the most efficient. There is a site that lists them in order https://www.energystar.gov/most-efficient/me-certified-clothes-washers/results . They don’t show all the 24″ models, and they work much better in an apartment for size. I would look at Bosch, Blomberg, and Miele. They really make very efficient washers.
They only make a downdraft range combo with the vent in the center. Jenn Air and Kitchen Aid make this option. If you 100% want it in the rear, then you can buy any slide in, and as long as you have the extra depth, brands like Elica Best and Zephyr make downdrafts specially designed to go behind a range. The issue is the motor will have to go behind it or in a basement remotely mounted, so the setup is more complicated to achieve and usually needs a much deeper cutout.
When you say cooling mode do you mean air fluff or air dry ? stuck on that cycle. If it is simply not heating on any cycle prob a bad igniter if gas or a bad element if electric. IF a electronic control panel you can unplug it for a few hours and see of it resets.
You paid attention to my posts and I appreciate that. I do love the 800. But with no dispenser, the 500 is decent. But to be honest, the GE Profile is a better unit. I get you need to stay in the budget, so even the baseline GE will be good. I have had some hiccups with the Bosch 500 lately. So I have kind of moved it off my list till I see them get corrected.
I have seen that when the outer coating de-laminates and usually requires a replacement part, unfortunately. No way to re-glue it as it is done at the factory.