It's close enough to work but not ideal, and it's a good idea to use a chain rated for 8 speeds.

It really depends on the wheel if a 7-speed wheel can go up to 10. There's some MTB hubs that had narrow freehubs for 7-speed. Freewheel hubs should really only be used up to 7 speeds. Some wheels put a 7 speed cassette on a 8-9-10 freehub with an adapter spacer. I couldn't tell you what your 7 speed wheel is.

You might be able to get away just changing the shifter, the chain and the cassette to 10 speed. These have to be purchased at once.

I'm not familiar with the parts you're talking about though. Spectro seems to be an internal gear hub system which is different from a derailer gear system, in which case you would have to change everything.