Right now (unless I'm missing something) the only way to play coop with randoms is by using the Host/Quick Join options in the main menu and then either they join your session or you theirs in the saloon.
This in my opinion is way too inaccessible for a game like this and I'd say to have the option to switch between solo/private/coop when you start a run would be very benefitial.

Let's say you are in your Saloon and from there you select your mission, you can switch before you start the mission between public, private and solo and then depending on your choice the game will either start you solo or try to match you with players that also start the mission (if it can't find anyone it will put you in Solo and someone can join your in progress mission up to a certain point). Once the mission is over every player can select to stay with the group or leave and get back to their own hub.
A system like this is used in many games that have a focus on Coop, take for example Warframe, The Division, Warhammer Vermintide/Darktide, and so on as it makes Coop far more easy to access and use and "removes" the barrier of feeling like you have to stick with a group once you were put into a match with them which some player do not like.

Edit:
I will also add, if a player leaves during a coop run they should be replaced by a bot. Right now their character remains but stays at the position they disconnected at (or when the area/floor is cleared fully and you move on they teleport once but then are stuck in place again).

Not sure if that's something that will be possible or not (leave it kind of open), will share it in any case with the team. Also, would like to mention that you don't have like okay I select X mission and then do it, in which I understand why this system is more important.

Yes, right now you can invite people (I will say this is more when you know the player) and then the search function. If not mistaken the team is working on including some kind of lobby code, so it is easier when sharing on discord or other socials.

For the latest part you mentions, this is how is currently working, if you are playing co-op and one of the person leaves (not the Host) the AI/Bot will replace it.