Must-Sees in Prague on a Tight Schedule? LGBT friendly?

Olomouc is lovely. It's a university town (though students were not in evidence during my mid-summer visit in 2018), so I wouldn't expect it to be particularly conservative.

Cesky Krumlov (which I have not visited) is said to be terribly touristy. The usual advice is to arrive late in the afternoon, spend the night, and depart before noon the next day. That allows a traveler to see it at its (current) best. Taking a day-trip there would be exactly the opposite. Frankly, I wouldn't do it. However, I'm retired and have the freedom to spend as much time as I want in Europe.

First off your concern about sharing a room with another female is unwarranted as many women travel with their friends ( I’m married to a man but take trips often with my female friend , and it has literally never occurred to me to be concerned about what anyone thought ) , perhaps if expressing open affection in public you might get a few raised eyebrows but Europe in general is “ live and let live “ when it comes to other people’s love lives .

And Prague - we stayed four Nights and I wish we had only stayed two . We didn’t do daytrip a outside city which would have made four nights more sensible . So yeah I think four nights with one or two day trips is ideal .
Prague is not very large , and it’s not like London or Paris with dozens of magnificent museums and monuments . 2-3 days is more than enough in my opinion.

As I recall, Cesky Krumlov is a pretty long ride from Prague. A friend who did it as a day trip told me that the amount of time being there wasn’t long enough, compared to what it took to get there. We stopped there on our way from Vienna, and stayed one night before continuing on to Prague.
One “must see” in Prague is the Municipal House. It’s a simply gorgeous Art Nouveau extravaganza. You need to reserve a place on one of the guided tours. The tours in English were fairly frequent and not very expensive.

I agree with Rosalyn about the glories of Municipal House (Obecni Dum) for those interested in Art Nouveau architecture. I highly recommend paying the modest additional fee for taking photos. I regretted not doing that, because the architectural details were beautiful.

I thought the Museum of Communism was very well done. It's not a super-quick visit, though.

I can not speak to the National Park, as we have not been there. We spent two nights Cesky Krumlov, arriving late in the afternoon and spending the next day there. For us, it was a huge disappointment and I wish we had not wasted precious vacation time there.
Lots of kitschy souvenir shops, mediocre food and sights not worth visiting. With that said, I can not imagine taking time from Prague as a day trip, which is one of my favorite cities in Europe. If you are interested in history, I would recommend that you check out Pavel Batel's website as he offers several private tours that are very reasonably priced. https://terezin-private-tours.com/general-prague-tour.html

My 2 cents worth, I’ve been to Czesky Krumlov, but not to Olomouc. And that was in over 20 years ago, when you would’ve been celebrating your 4th anniversary. Things may well have changed, touristically, but C.K. was really worth the short stay. What time of year are you going? Perhaps some times are less crowded than others.

We’d been on a bicycle trip along the Danube in Austria, then took a train towards Prague. The train engine on the Austrian side was fast; once we crossed border into the Czech Republic, the Austrian engine was uncoupled, and a much slower Czech engine was attached. The going was slower then, but we kept moving. I’m guessing all that equipment has been replaced, maybe several times, and the speeds are higher all around. And the journey really wasn’t that long, as I recall. But we stopped in C.K. for one night, as planned. We arrived just before noon, and the walk from the station into the town was a bit long. Taxis or some kind of shuttle system may exist now.

Seeing the sloping main square, which was actually a great place, was surprising, as it slopes to one end, not flat as most main stairs would seem to be. The Castle, complete with climbable, fairy-tale tower, wonderful (then) gardens, and ballroom and other spaces was a highlight. So was the Egon Schiele museum. And dinner outside, under the stars, was wonderful and delicious. If things have been cheapened to sucker visitors now, that would be a shame, and would hope that at least some places have kept decent standards. Stay the night - with the castle lit up, spend one night away from Prague, for a different experience. We departed by train mid-morning the next day, on to Prague.

In Prague, attend a concert - Mozart or otherwise. Even with just 3 days, you’ll be able to have a decent stay, without having cheated yourselves.

Slav Epic will be returning to Moravsky Krumlov this summer... but Mucha Museum in Prague has a video showing some of it and was a "must see" for me.
(We saw the Slav Epic exhibit in Prague a few years ago before it went on tour )
C.K. can be an overnight between Prague and Vienna using CK Shuttle for easy hotel-to-hotel transport. Just try to avoid the bachelorette "hen parties"

Look at prague4gay.com. It's run by a local guy called Petr. Extremely knowledgeable about the city and he runs tours aligned to LGBT experiences. He might be able to answer about Olomouc as well. If you go to CK then also check Student Agency bus schedules. Trip time is the same but nice to get the train down and the bus back for different views etc. Happy 25th, you'll have great fun in either place.

Old Town Square, Wenceslaus Square, Charles Bridge, Prague Castle, St. Vitus Cathedral, Jewish Cemetery.

A good YouTube channel to check out is "Honest Guide". It has some great tips on Prague and the videos are usually short and interesting as well as entertaining.