When I try to enter the mirror world map on the battlefield, it says "You started researching the ways to this world mirror (Progress: 0/5000)" at the top of the map. I'm guessing that this means I have to finish researching before I can enter the map? How long does that take?

You are correct.

In order to see how long it's going to be, you need to check each of your squad's pets' pages and check their value for "Research Potential". Add the research potential of all 4 pets together, that is the value added hourly to your current progress. According to birdlove's statement in a similar post, the exact value can vary up to 50% every hour. That way you can estimte how much longer it's going to be.

For the future, you can influence your pets' research potential by doing certain things - hover the mouse over the question mark next to Research Potential on their profile and click the Details link in the window that pops up. This also gives you a breakdown why a pet has as many research points as it does. 

Thank you! I had seen the research potential category on theor profiles before but didn't know what it was for--it was added after I went on hiatus several months ago so I'd never had to use it before. But gah, it's going to take my squad at least 2-3 days to research this map--I didn't realize it would take that long!