How much play in Shimano gear cassette? Chain stretch?

I have a small amount of play in my gear cassette. It wiggles back and forth maybe 1/16". Is this normal? Also, I'm putting quite a few miles on the bike. How much play should there be in the chain, and how can I tell if it needs tightened? Last, the gears are dusty on the inside, the opposite side of the chain. Is it ok to spray water from a hose on this area, as it's very difficult to reach, or just leave it alone?
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If the wiggle is transverse to the axle, no this is not normal. Is this the Shimano 7 speed sprocket cluster? They are bad to have the internal race back off and 1. wiggle 2. let the balls fall out on the road. *****y cheap design. I had to push the bike home 4 miles, good thing I wasn't 30 miles out.
The fix is an axle with a fixed race, made out of over size stock so it costs a dollar more. I think the brand was "we the people" on but I don't see it in stock today. Was $35 when I saw it 2 years ago. I bought a 2nd nut to tighten against the race to lock it, but it took 3 boxes from thebikeshopstore at $8 a pop to get the actual correct 3/8"x26 tpi nuts. Standard fine thread nuts are 3/8"x24; don't fit. Those people don't measure their parts, they just stick random garbage in baggies with the right name on them. It was easier to order a $10 tap from and make my own **** nut.
Chain wear is measured with a park gauge from same shop. If you have a dial or vernier caliper, the nominal chain pitch is 0.500". Worn chain stretches.
No you don't spray water or any solvent on the chain or sprocket. If you are oiling the chain often enough, the dirt will make a paste that looks like grease below the tips of the sprockets. You scrape this off with a screwdriver & throw it away. No paste, you're not oiling the chain often enough. I oil mine twice a month with 5 W ATF Type F or A (not dexron/mercon), or SUS32 generic hydraulic fluid (not JD NewHolland Case specification) from the farm store. The wrong oils have detergent, collect water out of the air. Spray waxes don't get inside the chain to lubricate the internal sleeve pin & side bearing. I use a pump oiler, and yes it drips off. Use a piece of packing paper to protect the floor. Use a binder clip on your pants to keep it off the oily chain.
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"If the wiggle is transverse to the axle, no this is not normal. Is this the Shimano 7 speed sprocket cluster?"

Thanks for the detailed reply. The rear derailleur is listed as a SHIMANO SLX 10 RD-M670 10S. It has 10 gears. The bike is a Gepida.
I'm not sure what you mean be "traverse." Facing the gear cassette, and pulling it towards me, I can feel some play.

Regarding the oil I am not oiling enough. I am using a spray lube. It stays quite clean. I have almost no paste build-up. The gears operate smoothly without squeaking, and I lube them more frequently than you do. What you're doing sounds like a big mess, with ATF flinging all over when you first oil it, but probably quite effective.
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No paste, you're not oiling the chain often enough.
This is not usually recommended. Such a paste contains dirt, which shortens the life of your chain. Recommended procedure is to wipe the chain after oiling to remove outside oils - what's important is that the inside joints are oiled - anything on the outside just holds dirt.

Here's a video from Park Tool on cleaning and oiling bicycle chains and drivetrains. Yes, it's pushes their products, but you can substitute whatever you want:

There is no "chain tightening" procedure. If your chain has stretched, then it needs to be replaced. The video shows a chain stretch measuring tool. You can buy one of those pretty cheap. Some people use an accurate ruler to measure (each link is exactly 1" long, so measure 10" of chain and be sure the location of the last rivet is within 1/16" of the first rivet), but mid-drive eBikes put a lot of stress on chains, so I think a relatively cheap chain stretch measuring tool is worth it for us.
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Very cool video. Love the chain cleaning tool, but as one comment mentioned there's probably a biodegradable orange oil substitute for solvent. I'll have to look for these supplies. Great tip on measuring the chain. Thanks.
UPDATE: Took my Gepida to another bike shop. The last one didn't do a very good job. The bike came back like new. The wiggle in the gear cassette was tightened. The brakes seem better than the last ones. The brake levers are correctly tightened now.