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.
You have a freewheel not a freehub. Therefore your wheel cannot take a cassette. Instead it has to have the sort where the clicky part is part of the cogs/block/cluster.
7 speed is already pushing the limits of the design, and bent axles are common. While 8 speed freehubs existed, they bent axles very easily and were a technical failure.
If you want 10 speed on that bike, you need a new rear wheel hub, which may work out more expensive than a complete new rear wheel. You also need a new shifter on the bars, a new 10 speed chain, and maybe a new derailleur especially if you get more teeth on the big cog.
Honestly this is a very expensive way to do it. Since you enjoy riding, consider a second bike. Then you have two bikes!
A quick search in my internet LBS returns a 10-speed freewheel that is 41.1mm wide, weights 624g and has a range between 11 and 34 cogs. If your current freewheel is 34 mm check if:
a) there is room for those extra 7.1mm
b) your current derailleur has enough move to reach all sprockets
c) you can get an indexed shifter for 10 speeds.
Is your frame steel, aluminium or carbon? The first one can be slightly set out to accommodate wider hub (you can put some spacers on the axle).
Making a step back with your question - what are you planning to achieve with this replacement? Wider range of gears? Or simply more speeds? For the latter you might want to read the question Modern road bike 8 spd vs 9 spd - what are benefits and is it just about the speed?
For the former you might want to stick with 7 speed freewheel only with 13-34 range?