I'm wondering if these destinations are feasible to do in a week.

1. Universal Orlando (is two days enough to see everything?)
2. Everglades Nat'l Park (again, is two days enough to soak it in?)
3. Panthers hockey game (they play in Sunrise, which seems to be a suburb of Fort Lauderdale on the map).

Would Miami work as a base city for the Everglades, or is it too far?

Is two days each, enough time for both Universal Orlando and for the Everglades?
Yes.
Both do-able in a week?
Yes.

Will it mean a lot of rushing around?
Unfortunately, yes.
You can leave Orlando first thing in the morning, get to Fort Lauderdale in the afternoon, and see a Panthers game that evening, but it won't be a relaxed day.

Miami is BARELY adequate for day trips to the Everglades, but it will require two to three hours each way between the two. If ENP is a central focus of this trip, I'd stay closer to the park, like in Homestead.

More specifically, Fort Lauderdale to ENP and back in a day would mean more time fighting traffic than seeing gators.

Thanks for the great response, Paul. Maybe we'll do a night in Fort Lauderdale then before driving to a location closer to ENP.

If you are flying, do do an open jaw flight, I would not use a single base since your destinations are not close to each other. Think about begining in Olando, then on to ENP and wind up in Ft. Lauderdale or one of the communities at bit north or south.

Update:

The trip is booked! We're flying into Orlando and staying across the street from the Universal gate. We will then rent a car and drive to Florida City, where we have another hotel that we're using as a base for the Everglades N.P.. The Panthers game was taken out of the trip, and instead we're going to take a day trip down to Key Largo. Then, we fly back home from Fort Lauderdale (about $50 per person cheaper than Miami).