Authy had a breach^{in 2022 as far as I remember}, is closed-source and doesn't allow you easily to export your secrets. I'm moving away from Authy...

For Android I recommend FreeOTP, Aegis or 2FAS. I'm no iOS user, but Raivo seems a good choice for this platform (no personal experience, DYOR).

I don't think it's a good idea to make your mobile phone the main root of all your data. It's easy to loose and likely prone to be stolen for the more valuable devices. Avoid receiving OTP via SMS or similar.

My internet connection at home isn't related in any way with my cellphone number. Mobile data in cellphone network is related to it, but I don't quite get your point with it.