Yes Sunrise and Paradise have different entries with different parking lots so you need different permits. It's all about trying to minimize waits at the booth and making sure you'll have a spot to park.

With that amount of time I'd spend 3 days in Seattle, 3 days in or around Mt Rainier.

As far as which order you might want to check where you can find a place to stay. Guessing more options for Seattle hotels over a weekend. So Fri/Sat/Sun in Seattle, Mon/Tues/Wed at Rainier area, fly out Thursday night.

You definitely don't have time to add ONP, unless you want to skip everything else.

As far as teen girls in Seattle... yes what are her interests? There's MoPop (contempory music museum), places to go up high for a city view (Space Needle, Sky View Observatory, Smith Tower). The Underground Tour and Klondike Museum (gold rush) in Pioneer Square historic district. Of course there is shopping and fashion (flagship Nordstrom store), art galleries, boutiques, spice shops, bakeries, a gazillion coffee shops, ice cream... Do not miss the Pike Place Market, both for gift shopping and places to eat.