If I want to hold other coins, (Litecoin) do I need to sign up to an exchange?
for holding them, no. you sign up on exchanges to buy these altcoins. each of them should have their own wallet. visit their respective website or announcement on bitcointalk to find their official wallets (example: litecoin.org)Quote
do I need a different wallet? (current Trezor)
yes but some wallets like Trezon support multiple cryptocurrencies. if i am not mistaken Trezor supports Litecoin and a couple of others, check out their website for details on how.Quote
and can I sign up for an exchange outside current Country?
yes. most of them allow that. if they block your country, they will surely give you a warning when you are signing up. 