Will I lose progress if I unlock items and bases before starting the first mission?

After completing only two or three missions, I went through and liberated every single province in the game.

Now playing the missions all the bases, vehicles, and equipment I had liberated / unlocked are still available.

This actually makes some of the missions extremely easy, as instead of having to drive a tank through enemy territory, it is a relatively easy drive from beginning to end.

You should not lose guns or upgrades.

However, if you haven't unlocked the wingsuit yet (which I am pretty sure unlocks during the tutorial), there is still a copious amount of gameplay for you to complete. There are three islands, and the first part of the game is only on the first.

Source: 200+ hours in JC3

No, but I prefer to follow the missions. The three cetrecom bases all have some sort of weapon (air strike, Bavarium nuke, and something else) that can be removed by doing the missions. Falcon Maxine is nearly impossible to liberate without doing this, as it launches a bavarium NUKE at you every 10 seconds or so. This coupled with the massive size of the base, makes it insanely hard to liberate. So I recommend doing the missions first. You do you though, your progress will still be saved.