I haven't done inter-terminal transfer at TPE so I couldn't tell you anything about it. Just ask SQ lounge or airport staff if there's additional screening for transfers.

If you don't exit the immigration then you won't even be able to use the TG lounge since it's on landside in Terminal 1. So yeah try to stay in the SQ lounge. The food offerings in the SQ lounge is awesome (well at least when I was there) - Chinese snacks, bums, fried rice... and Haagen-Daaz ice cream! I

SQ lounge staff in TPE are friendly, unlike those in SYD who give you a look when you don't follow their "advice" to use UA or NZ lounges there.