Is it possible to use Trezor Safe 3/5 with only third-party wallets, without ever using Trezor Suite? Will I be able to use multiple accounts and/or multiple cryptocurrencies with a single device?
Yes you can.
About your second question, yes you can use single device for multiple crypto (whether you want to use trezor suite / 3rd party wallets).
About multiple accounts, i am not sure, maybe you mean you want to generate separated wallets with same device using Trezor’s additional passphrase?