Potions add toxicity to slaves, which can be monitored using the Mind Read spell. When that toxicity is above 60 you will have a chance of mutations, and need to wait until it dissipates. Each potion has it's own amount of toxicity and the only resource for knowing how much is the items.gd script in the game's files. These values range from 0 for Hair Dye to 50 for an Elixir of Clarity.
Not really certain what you are asking for in your second post. The age of a slave has almost no impact on them at all, so there are no age requirements nor patterns for Specializations. Children, as in offspring from a pregnant person, will not inherit a specialization from their parents so they will not start with one.