Does the speed at which you reach the highest temperature affect salt absorption? Does slow cooking matter before or after reaching the highest temperature or both?
Cooking time is the main thing that affects how anything liquid/dissolved works its way into a food. If you're actually slow cooking, the vast majority of the time is after you've reached a stable temperature, so your question doesn't really matter much. Further, if you slow cook long enough, you can just shred it (tear it apart) and the liquid will quickly have soaked all the way through it.
If you're cooking for a short enough time that the full temperature period is comparable to the heating up period, then yes, things may happen slightly faster at full temperature. When things are still cold, it's like you're trying to marinate the food. But there are no big magical effects; nothing weird will happen if you bring something slowly or quickly up to temperature.