If you're traveling by vehicle, keeping a portable kitchen isn't a big deal. But this one is for those out there living out of suitcases for long periods — your truckers, flight crew, and just plain those of us that would rather not own a car.
1) Refrigeration
Until they actually invent the Mr. Freeze Gun, refrigeration is entirely a procure-on-site situation.
Obviously, you want a place that has at least a mini-fridge. A lot of lower-end hotels and motel chains are stepping up their game in this area as part of a general strategy of upgrading and improvement. But there are still a lot of discounts deals out there that don't offer them, and it's also pretty common in cheaper AirBnBs (or it's a shared fridge situation where anything good you leave stands to get poached).
The cheapest of the little cube mini-fridges are usually around $70 and the good-sized ones with the little freezer compartment are usually around $150, so that's probably not a buy-then-sell option unless you're sure you're going to be there for months.
Unfortunately, there isn't really a good alternate solution. The only viable one is to set up a cooler with dry ice, but you'll probably need to add new dry ice once per day or two and you'll be limited to keeping all your food frozen. You also need to keep it in a well-ventilated place as the dry ice will create oxygen-displacing CO2 as it sublimates. This works best at a hotel or motel that has a free ice machine to take advantage of.
If you want to go that route - pick up a cheap Styrofoam cooler from any local store, Family Dollar or pretty much any big box that sells food have them for somewhere under $5. While you're at it, pick up a good amount of those free newspapers that are out front of stores everywhere. Either bring the cooler to where you're buying the dry ice, or ideally buy them both in the same place (Walmart is the most reliable choice to have both of these things at a low price ... grocery stores and gas stations frequently have the dry ice but not the cheap cooler). Put the dry ice in the cooler ASAP as it will start sublimating as soon as they take it out of cold storage, and get it back home ASAP. You have a few hours at room temp before it will totally evaporate, but the longer you take the less time it will work in your cooler.
Put the dry ice on the bottom of the cooler and layer some newspaper over it. Then get some regular ice in the cooler. Fill up any remaining space with newspaper (the dry ice will keep longer the less space there is in the cooler). You can then displace newspaper to add food as needed.
Not the best solution, and works best if you are staying near a dry ice source as you'll probably need to replace it every 24-36 hours or so. But it's an option.
2) Single Burner Stove Top
Electric is the way to go here if you're staying indoors with electricity. There aren't a lot of "mini" models I could find listed online ... probably low demand, and the fact that you would want to let a pot or pan sit for a while first to get more even heating. But I did find this unusually small model by a company called Eastbuy for around $30.
A regular single-burner electric cooker is likely small and flat enough to be workable in a suitcase, though. This model on Amazon is around $20 and looks like it has a smaller profile than usual than the ones you see in the big box stores.
If you're in a "no electricity" situation like a rustic cabin or somesuch, I guess you could try Sterno (canned heat). These are the slow-burning cans of fuel that are used to keep buffet trays hot. You can buy a stove kit for around $20-25 and then just need to swap in cans of fuel as needed (usually around $5 and give you 2-3 hours of cooking time each). The only limitations here are that these don't get as hot or heat up as quick as an electric range would (it can cook meat but you want to slice it as thin as possible), and there will be fumes so you need to be in a ventilated area.
3) Pot And Pan
You might be able to swing a small skillet in a suitcase (I highly recommend Farberware, best cheap non-stick skillet I've owned in my life), but a cookpot is just a big lumpy space-eater that's usually unmanageable.
The answer? Collapsible flat-folding camping cookpot! You can even fold this thing down slightly to cook eggs and meat and such to eliminate the skillet, but it's tricky with anything greasy or liquidy that needs flipping.
4) Coffee Brewing
There are a few different ways to approach this vital necessity of life. The one I DON'T recommend is relying on hotel room coffee makers, which are often either unhygenic or make coffee that tastes like burnt plastic because they should have been replaced years ago.
The one I most recommend is a "mini" or "single serve" French press. Ideally you can go to the grocery store and get a coarse grind on fresh beans, but if you need to use standard canned pre-packaged coffee it's fine too, it's just a little more annoying to clean the fine grounds out of the plunger completely.
If you want to procure everything on site for about $3, go to the local dollar store and grab a large cup for brewing in, a coffee mug and some of the really large brew basket paper filters. Put the grounds in the cup and pour the hot water directly in with them. Line the mug with one of the brew basket filters, a good amount should be sticking out the top so that you can pull it out. Let the coffee brew in the cup for about five minutes, pour into the mug not worrying much about grounds getting in, pull out the filter slowly and carefully to extract the grounds that slipped through without sloshing, voila. Alternately, strap the filter to the top of the cup with a rubber band and pour through it.
For boiling water you can either use the range top + pot, or if you have space for a more elegant solution just get a mini electric kettle.
5) Bowl and Plate
A plate is no big deal. Grab one from the dollar store for $1 (or a pack of paper or plastic plates for the same). Or you can invest in a sturdy, thin plastic plate to pack with you and re-use.
Bowls are another space-taking thing. You could grab one for $1 as well, but an alternative is to bring a collapsible camping bowl.
6) Utensils
The basics aren't really a big deal, but if you want to save space there are folding sets, folding spatulas / tongs, and folding ladles.