It is not possible to install 10-speed cassettes on 7-speed hubs, the reason is that 7-speed hubs are old, and traditional freewheel hubs have their specific 7-speed cassettes, but 10-speed cassettes have their specific multi-cog hubs, which are compatible with 8,9,10 and 11-speed cassettes, there is also different ion the length and diameter of the 7-speed and 10-speed hubs, due to which their specific cassettes are not swapped and interchanged with each other. although there are many 7-speed bikes that have compatible and the same space between the left and right dropouts, even if the dropouts are the same, you cannot swap or interchange their cassettes and hubs, the reason is that the rear wheels of 10-speed cassette and 7-speed hubs are different from each other, and it will affect much more in shifting of gears between the cogs of the cassette, in simple words we can say as their swapping and interchanging is not possible.
New Rear Wheel:
This setup requires a new rear wheel, which is usually much more expensive, if you support this setup or if you are building and making your own bike, then this setup is beneficial for you, but if you have an old sued bike, then it would not be better for you because there are also other things which do not work between this setup.
The new wheel also needs re-dishing and re-lacing of spokes and spoke nipples, if you are a mechanic then your bucks will save, but if you are a beginner then you have to spend almost 100$ to 200$ on re-dishing and re-lacing of the wheel.
Shifters:
7-speed shifters are limited to 7 gears but 10-speed cassettes require multi-cog or 10-speed shifters, it shows that previous 7-speed shifters are totally useless in this setup and you have to replace them with the new 10-speed shifters, you are going to spend furthermore bucks on these shifters.
Rear Derailleur:
The rear Derailleurs of 7-speed bikes have their specific and limited range of motion but 10-speed bikes have a good and excellent range of motion, so if you are installing a 10-speed cassette on a 7-speed hub then it is a must to replace or interchange your rear derailleur which is surely expensive and needs a lot of money.
Chain Rings Of 7-Speed And 10-Speed Bikes:
Chain rings of 7-speed bikes are wider as compared to 10-speed bikes, which means the width and diameter of the teeth of the cogs are different from each other, and also the spacing between one tooth to the next is also different from each other, which shows that 7-speed chains are wider and 10-speed chains are narrower, due to which chains of both of these different number of bikes are not compatible and are not swapped with each other, you also need to buy 10-speed cassette chain for 10-speed cassette, and also you have to replace 7-speed chain rings with 10-speed chainrings.
Difference Of Dropout Spacing:
There is a common difference between the spacing of the dropouts between mountain and road bikes, all categories of mountain bikes have the same spacing which is recorded as 135millimeters, but if you are a road biker then there is a little bit of difficulty for your in spacing, you have to face small difference in dropouts, 7-speed road bikes have 126millimters of dropout spacing and 10-speed road bikes have 130millimeters of spacing, there is a clear cut difference of 4 millimeters, you have to face this 4mm difference in this 7-speed hub and 10-speed cassette difference, this difference does not affect hub, cassette, or anything but it will widely affect rear wheel of your bike, wheels of 7-speed and 10-speed bikes are made differently based on this dropout spacing difference.
When you install the rear wheel then you will notice that dropouts of your bike frame are too far and affecting the exact location of the rear wheel and drivetrain, we have one way for this issue, if your bike frame is of aluminum and carbon material then it is not possible to bring the dropouts of bike frame 4mm inward and nearer with each other, because these frames would break easily due to less stretching and strengthening power, but if your bike frame is of steel material then, this material is famous for more strength and strength without affecting the joints of bike frame, you can easily cover this small difference on steel frames.
Length Of Rear Axle:
&-speed bikes have their specific length of axles, but 10-speed bikes have different length axles, and when you are installing a 10-speed cassette on a 7-speed hub then you have made the length of your hub axle compatible with the cassette and dropouts, it is important to locate the rear wheel and the ratchets of the hub on a specific place within the drive train.
Spacers:
7-speed hub and 10-speed cassette would also be compatible with each other by inserting spacers on the drive and non-drive side of the hub axle, you have to buy accurately sized spacers so that the diameter and thickness of spacers should exactly match the hub axle, these spacers would help a lot in reducing the spacing of the dropouts and also maintaining the rear wheel.