Can quests disappear before I do them

Go to a shop, and then go to the weapon you have (or want to buy), and then select the 'Equip' menu:

Then, assign that weapon to a slot:

You should now be able to switch between all four weapons, assuming you have assigned weapons to all four slots and have the Heavy Duty Holster (four weapon slots).

Fully Inked is an achievement that requires you to "Earn every tattoo by learning all the skills". This confirms that it is possible to earn all skills and since there is no new game + mode, it must be possible to earn on a single playthrough.

It is unlikely that all story and side missions combined would provide enough XP to unlock all skills however given the enormous amount of enemies populating the game world, it is very likely that you could pick up any remaining skill points by grinding enemies.

As other posters have mentioned, enemies respawn in the areas around an enemy-controlled outpost (red bubbles on your map). You could clear these areas multiple times (as long as you don't actually capture the outpost) OR you could maximize the amount of XP earned from each enemy you encounter by performing headshots (2.5x XP) or special melee takedowns (3x - 7.5x XP).

If you are set on unlocking all 24 skills as soon as possible I would make unlocking the Death From Above and Death From Below skills a priority. These skills allow you to perform melee takedowns from above or below an enemy for a 5x XP bonus. There are also upgraded 'Dual' versions of each of these skills which allow you to subdue 2 enemies at once (7.5x XP). Getting these skills early and using the takedowns often could significantly shorten the amount of enemies required to earn all skill points necessary.