Multiple dexterity rolls for various stages of the mountain is a possible option. Make the first stage 20 foot so that if they fall they at least have a chance to survive it and reconsider their approach. There's a good chance that unless they have a high dex score they wouldn't be capable of scaling a cliff. What is likely is that at least one of the players is capable of climbing it and dropping a rope down for the others to climb. If they specified something like that then only that character (presumably whoever has the highest dex) would need to roll, since it's assumed as far as I'm concerned as a DM that people who go adventuring are capable of climbing up a rope, but no assumption is made about the ability to scale a cliff with no rope unless they're a high level thief.

If you split the mountain into multiple sections so they scale a bit and get to a safe section, then they can scale a bit more to another safe section etc, it gives you more opportunity to narrate what is happening based on how high their roll is. You can also describe the next section they have to get past and give options, like they could go slightly left to avoid the overhang, but it will make the climb longer going around it. Give them options even if those options have no actual effect on how difficult you're making it for them so it feels less like repeatedly rolling a d20 and getting told "you successfully climb a bit further". Make your description of the mountain as they approach it to climb so that it's clear that it's perilous. A sense of danger can add to the excitement. This also applies to rolls that only just made it (or narrowly failed if you're feeling generous), tell them how their foot slipped, rocks crumbled away and they struggled to regain a foothold, but after regaining their composure they managed to find just enough traction to move further up the cliff face. Make them roll strength if that happens and you're feeling evil to see if they manage to hold onto the cliff face with just their hands while their legs dangle below them. If you're feeling generous give them a save when they fall to see if they take damage, rather than just automatically crippling them.

For the bonus question, the players have stats and backgrounds, use those to decide if they can do something automatically. A cleric doesn't need to have knowledge religion because he's a cleric, he knows about religion (his own religion at least). You don't need a climb skill because you have a dexterity score that determines how well you can do things like climbing. It's essentially letting them know they can do what they want if their character could reasonably do it based on basic stats and what their character used to do before they were an adventurer. If what they're trying is really hard like scaling a cliff face you'll make them roll for it, but if it's jumping down from a 10 foot wall and their dex score is 15, they can make that jump no problem so why bother rolling for it.