Is 7-speed for 10-speed cassette replacement possible? I have a city/touring bike with Shimano MF-TZ21 freewheel hub which is 340mm wide (I measured). I would like to change it to a 10-speed cassette 11-25 or 11-28 but I am not sure if it can fit between my chainstays (129mm). I know I would need to buy shorter hub also (and deraileurs, shifters, chain etc.). In bike service they told me (without measurements) they are not sure if new 10-speed package can fit between my chainstay. I googled 10-speed Shimano cassette dimensions but I could hardly find anything I can interpret correctly.
Please help, can I fit new package in 129mm?
Higher number of speed cassettes and freewheel are not wider, they have smaller sprocket spacing.

If you were talking about a cassette/freehub system, you would be able to fit a 10 speed cassette on the existing hub because the freehub body is the same width for 7 to 10 speed cassettes. (11 speed requires a wider freehub body on road bikes).

I'm not terribly familiar with freewheels, but I would think the same principle holds.

However, if you have a 7 speed rear, freewheel equipped, Tourney level bike, and you want a 10 speed rear, you really want a whole different bike. To get a 10 speed rear you need to upgrade to Tiagra or Deore, depending on if you have a road or mountain style bike. You'll need to replace the crankset as well as the derailleurs and shifters. Buying the bare drivetrain groupset will be expensive, require a lot of time to install, require special tools and will give you a lot of headaches trying to get it set up correctly. 

Go the easy route and get an upgraded bike. It will probably be cheaper and you'll get an upgrade on every component, not just the drivetrain and it will work right out of the box.