I have a Giant Anthem bike that has a 10-cog cassette wheel. And I have just purchased an e-bike hub motor that is a freewheel with seven cogs. Can I go ahead and fit it without any problems?
No, you would not be able to fit it without any problems. Your Giant Anthem is using an 8/9/10 speed compatible freehub, and your shifters are also matched to the 10-speed cassette there. Even if your e-bike hub motor with 7-cogs could physically fit your Anthem, you will not be able to make the 10-speed shifter work with the 7-speed hub.
If the e-bike hub could physically fit, you would need to change out your chain (a 10-speed chain is too narrow), your rear shifter (it may be difficult to impossible to find a 7-speed shifter that will match your front shifter), and your rear derailleur would also likely not be compatible (I don't recall if a 10 speed RD has the same cable pull ratio as a 7/8/9 pull ratio).
If it is a true 7-speed freehub, then you would not be able to mount your 10-speed cassette on it instead of the 7 speed cassette (a 7 speed freehub is too narrow).
Yes, there would be problems.