Is all of this legal?
The emulator program is a free trial (30 days). After this, please purchase the program if you want to continue using it. Alternatively, there are other emulator programs available, some of which are free.
The Spectrum computer can be emulated legally, as Amstrad have given permission for the Spectrum ROMs to be used with emulator programs.
The games themselves are mostly 'unknown'. Some publishers have given permission, a few have not, but the vast majority are unknown, as the companies which hold the copyright are often no longer in business, or were taken over/merged with various other organisations.
Further details on copyrights is available on the World of Spectrum site.
Using real cassettes
If you have some Spectrum games on real cassettes left over from the 1980s, these can also be used with the emulator. You need to connect a cassette player to your PC, set the required options in the emaulator and then you can play the cassette just as you would with a real Spectrum. The same problems apply regarding volume levels, interference, damaged cassettes and so on, only today, the problems will be much worse as the cassettes are likely to be 20 years old.
Further info
The Spectaculator emulator is a 30 day trial, but the full version is a one-time payment of less than 20GBP. This is well worth it, as buying a manky old Spectrum of unknown quality on eBay will cost more than that.
If you really want an actual Spectrum, it is far better looking at local car boot (garage) sales or in the local newspaper classifieds. Despite sellers claiming that 'mint' Spectrums are rare, just remember that millions were sold in the UK alone, and there are still huge numbers stuck in lofts, cupboards and the like. Plenty of these had minimal use as they were purchased at Christmas and the novelty wore off by mid January!