Does job class matter for Retainers?

[Zetsumei_Tsunarashi]

The most important thing to remember when deciding your retainer class imo, is that your retainer can not level up past your level in a given class. For example if you have ninja leveled up to say level 45, then the highest level your retainer can be as a ninja is level 45. That includes if you buy one of the retainer level skip potions. So even though the level skip potion says it will level up the retainer to level 60, it would only level up to level 45 after using the potion.

Long story short, make sure you pick a class for your retainer that is already leveled up as much as possible so you don't get stuck having to level a new job or reset your retainer's levels to fix the problem.

Can you actually have a cooking retainer? (never tried) And if so - does it make any sense to have a crafter retainer?

Nope, you can't make a crafting retainer. DoW/M or DoL only.

I am considering to get the companion app and receive a free retainer.

I would highly advise against this unless you are planning to actually utilize the apps features on a regular basis (I don't know why you would because the app is garbage). The retainer is far from "free". You only get the retainer with the premium subscription to the app which costs $5 a month. An additional retainer from the mogstation only costs $2 a month. So for the price you pay for the app, you could actually get 2 additional retainers instead of the 1 additional you would get from the app. So in reality, you're actually paying more than double the price for a retainer from the companion app. Additionally if you ever decided to stop paying the premium service fee for the app, you would lose access to that retainer. So you'd be stuck paying double the price for that retainer forever and undoing that mistake would mean leveling up a new retainer from scratch.