Regarding support, IBM documentation can be confusing so be sure to read carefully. Windows 7 is supported for 8.4.1 as client only! FM, Transformer, and client access to server are supported - as per the IBM document linked here up this thread. Windows 7 is NOT a supported environment for Cognos BI Server 8.4.1.
Having said that, I installed Cognos 8.4.1 on Windows 7 several times with much success, although there are some flakes and quirks. I will NOT recommend any of my clients to have their production environment based on Windows 7 (Because it is not supported and because it's not very stable). However, if it's a home installation for you (As it seems to be since you're using Win7 Home version) it should be fine.
As for 8.4, I never tried and it might be even flakier (Though I'm pretty certain it would work).