You can add a slimeball to your inventory in Survival mode by killing a slime. You can usually find a slime in a Swampland biome. Slimes will spawn in the dark, usually at night. If you are having trouble finding a slime, you can summon a slime using a cheat or you can use a spawn egg.
Then, can a villager climb a ladder?
Mobs do not use ladders or vines in their path-finding, but if they are pushed by something into a climbable, they will climb it.
Can a villager open a gate?
No, villagers are not able to use fence gates. I tested this in creative, and they always used a long route with a door rather than a short route with a fence gate. I even blocked off the door and they just stood there.
Can you breed villagers in Minecraft?
This will not cause them to immediately seek out a mate, however. Villagers can also become willing by having either 3 bread, 12 carrots or 12 potatoes in one stack in their inventory. You can also throw bread, carrots, beetroots, or potatoes at the villagers yourself to encourage breeding.