Is there a way to access the stat/skill levels of other actors besides the first?

Yes, in fact the correct way to get a skill would be to access the game actor first, omitting the actorId is just a shorter way to get the first actors stat or skill level. 

for example

// For stats
$sl.getStatLevel(actorId, 'statName');
// For skills
$sl.getSkillLevel(actorId, 'skillName');

and every member has to make a check of their "Climbing" skill. How would I go about implementing such checks?

You could do a for loop that  turns on or off a switch, something like this.

$gameParty.members().forEach(actor => {
    if ($sl.getStatLevel(actor.actorId(), 'climbing') < 10) {
        $gameSwitches.setValue(1, false);     return; 
    }  
    $gameSwitches.setValue(1, true);
});

Besides that, I noticed there is a Notetag for equipment to increase custom stats. Is there also a way for equipment to increase custom skill levels?

There is currently no way to manipulate skill levels with notetags, that may be something we look at for future updates.