Actually, Jake, I have not come across this Mei before, so I cannot answer all of your questions.
Sei-Shu refers to the province of Ise, which you can look up on a map. The famous Ise Jingu shrine is located there, an ancient focus for pilgrims from all over Japan.
Following the logic of Mei, Oishi refers to a village or town called O-ishi (Big Rock). It’s probably still there if you look on a map of Matsuzaka.
The list of gunsmiths shows ten starting with 青井 Ao+i (Blue well) so this must be the family business name, inherited or passed down. There may have been branches of this because two smiths are listed as Aoi Takezo, 1st and 2nd generation.
Your gun however, has Aoi Takezo plus this further name which confuses me as it is not listed. It is however listed for a smith called Aoi Chin Takanaga. I wonder if this is the same person. The Chin name looks somewhat Chinese to me, so I wonder if he was taken in and taught the business? Just my own imagination, however.
In a minute I'll post for you here the names section for Aoi smiths in Ise, Matsuyama, O-ishi. You can play with them.