Feats can be used simultaneously

The only requirement to use a feat is that you meet any prerequisite specified in the feat. No feats that have been released thus far have a requirement that "no other feat has been used this turn," so there's nothing preventing you from using both.

Mobile allows you to not provoke opportunity attacks from creatures you've attacked that round, and Crossbow Expert allows you to make a Bonus Action attack with a hand crossbow, if you've made an attack with a one-handed weapon.

What's not legal, as Daze413 pointed out, is that you're using your "free object interaction" more than once in a round, in order to switch between weapons, or to drop your shortsword and pick it back up multiple times. You cannot dual-wield in this situation, because crossbows need a free hand to be reloaded. If they were made in consecutive order, (sword, sword, crossbow, crossbow) it would be fine, however.

While it's hard to judge balance, especially when it's not clear what you're balancing against, these two house-rules both have a pretty effective point of comparison... in an older edition.

In v3.5 of Dungeons & Dragons, Wizards had the option to something like what you suggest. They could opt to prepare only part of their spells and keep the rest open, so they could be filled in later. Wizards were considered one of the most ridiculously powerful and flexible classes in the game.

What you are proposing is even more powerful, because the Wizard at least required at least 15 minutes of rest to prepare a new spell in an open slot.

In my games, I've seen this feature used by a few players. (Most opted to just prep everything at the start of the day.) Here's what happened:

• Gameplay slowed down. Whenever my players would run into a problem that they could conceivably solve half an hour later, the Wizard would go down his spellbook and look for possible solutions that they could prep. Wizards tend to have a lot of spells in there, so this would take a while. Lots of banter would ensue over which spell the Wizard might prep. Half the time, in the end no spell was prepped and a different solution was chosen altogether.

• Players would look to the Wizard even more than normal. A Wizard who preps everything at the start of the day is incredibly flexible. A Wizard who can prep anything once you know what you're up against is just unreasonably flexible.

• The Wizard would ultimately stop using the feature because it's a huge mental load. As the spellbook grows, the number of things a Wizard could do in a given situations rises very rapidly. It was doable for a few levels, but it quickly became too much.

In the end, most players chose not to use the ability because it was both too good and too cumbersome. It would take too much table-time away from having fun and made a lot of encounters too easy.

You are likely to experience the same things as level goes up; the caster's options will go up rapidly, which increases their cognitive load and makes them much more powerful. If you think picking what to do in combat is bad when you have 5 spells to pick from, imagine how bad it gets when you have your entire spell list to chose from.

One of the reasons 5e cut down on how many spells you can cast at a given moment is because combat flows a bit more smoothly that way. And, like I said in my comment, there is a reward in actually thinking ahead now that you will lose. It's not your fault (or problem) that your players aren't gathering information about what they will be facing.

(As for hounding players to select spells; I always rule that after a long rest you prep the same spells as the day before unless you tell me you want to make changes. Most players have a default loadout that they use most of the time anyway).