No 10-speed derailleur and 7-speed cassettes are not compatible with each other, the reason is that the 10-speed derailleur is compatible with the 10-speed cassettes, in this case, the cassette is 7-speed which means the gear shifters are also compatible with the 7-speed cassette, but these shifters are not compatible for different sized 10-speed derailleurs, the main difference is the length of the cables when you pull the levers of gears shifters bite ratio would also remain different and almost non-compatible with the 10-speed derailleurs.
You also have to measure the length of the chain by uninstalling the quick links of the chains, if it is a Shimano chain then buy a new quick link because old chain pins won’t work, if it is another branded chain then you can reuse chain pins and quick links. The 10-speed derailleur and 7-speed cassette need different lengths of chains and the previous 7-speed cassette chain is not compatible with the 10-speed derailleur. these derailleurs would also contain small teeth and small jockey wheels, these wheels and teeth are specific for specific chains, 10-speed derailleurs are non-compatible with the 7-speed cassette chain.
Width Of Chain:
Chain widths are different for 7-speed and 10-speed cassettes and derailleurs, so you have to look for another compatible width chain for 10-speed derailleurs with the 7-speed cassette. because 7-speed chains are also wider and thicker in diameter than the 10-speed chains and the 10-speed derailleur is compatible only with the 10-speed chain.
Chainring:
Chain rings of 7-speed cassettes are wider and 10-speed chains are narrower in diameter, so the chain is not compatible with the chain ring, either you have to replace the chain or chain ring for making the setup compatible with each other.
Difference In Gear Shifters:
10-speed derailleurs would only work with the friction shifters, but if your bike has indexed shifters, these won’t work with the derailleur due to the difference in cable pull ratio and the length of the cable.
Lining Of The Derailleur:
The derailleur won’t stay properly in an exact place and its lining will continuously change when you shift the gears from lower to higher and higher to lower cogs. it is also possible that the derailleur would suddenly cause the locking of the rear wheel and the whole rear wheel including the spoke and flange of the hub would ruin and destroy, so for this setup, you have to make each and everything compatible with the 10-speed derailleur and 7-speed cassette.
Irritating Noises:
After doing the exact setup you will hear large irritating noises arising from the rear wheel sides, everything is rubbing with each other which includes, chain derailleur, derailleur hanger, cogs, and teeth of the chain ring and cassette.
Gear Spacing On 7-Speed Cassette:
The cogs and sprockets of the 7-speed cassettes are not compatible with the 10-speed chain and when you change gears, the chain would jump badly into the other sprocket and cog. this jumping of chain on a 7-speed cassette is due to a non-compatible chain ad cassette.
Spacers:
Spacers are installed on the non-drive side of the rear wheel for correct chain lining, spacers would also adjust the lining of the 10-speed derailleurs, and also these spacers are inserted on both the drive and non-drive side of the bottom bracket of the crankset to make the chainrings and 7-speed cassettes compatible with each other.
Adjusting Levers:
You also have to adjust the lever bite and the lever pull ratio. because the ratio is different for 7-speed,8-speed,9-speed, and 10-speed levers, you have to learn a new skill of lever pulling, handling, and gripping for this new setup.
Number Of gears In The Gear Shifters:
The 10-speed derailleur has 10 gears and the 7-speed cassette has 7 gears in the gear shifters, you have to maintain the number of gears for the 10-speed derailleur and 7-speed cassette.
Difference between Mountain And Road Bike Setup:
Gear shifters of 10-speed derailleurs for mountain and rod bikes are not compatible with each other. so if you are doing the current setup for a road bike, you have to buy new road-compatible gear shifters, and if you are doing the same setup on a mountain bike, you have to buy new compatible mountain bike gear shifters for this setup. other categories of bikes like gravel biking and beach cruiser, it is not to use different categories of bike parts(derailleur, cassette, gear shifters) on different category bikes.
Conclusion:
10 -speed derailleurs are not installed with the 7-speed cassette and 7-speed chain, and also you need 10-speed compatible gear shifters for the same setup, this setup is totally non-compatible because there is a large difference between7-speed chain and the 10-speed chain.