Your take on the stats is *mostly* right with a few corrections:
- Dex is the main stat for ranged DPS classes and rogue/ninja. Pugilist is Strength
- Spell speed has a small effect on the damage/healing of DoTs and HoTs, not their tick speed (which is always the static once every 3 seconds)
- Skill speed reduces the GCD for physical classes, making it lower than the base 2.5 seconds. This is also true of spell speed for casters, which is reflected in their reduced cast times as well.

Materia on your leveling gear will provide mostly negligible benefits to you, especially with stats that have breakpoints like spell speed. Stat priorities haven't been determined yet this expansion to my knowledge, but given last expansion, something like this will be the case:
- Mind (and weapon damage) is your most important stat, but you cannot meld it.
- Determination is linear, while crit is exponential and affects both rate and percentage increase when you get a crit. Det will be smaller and more consistent, while crit will quickly overtake it in usefulness
- Spell speed will provide a benefit in the same ballpark as Det, but it has breakpoints where you won't get any faster unless you have a certain amount.
- Piety is okay, but won't help much unless you stack a lot of it. At present, White Mage will need it the least, Astrologian the most.

Food is much better for the experience boost than the stats. They won't give you a ton of noticeable benefit. Food that follows your stat priorities is best at endgame.

Bonus answer: all three healers get HoTs. White Mage has some beginning with a single target around level 35 and an AOE at 50, Scholar gets them from their fairy, and Astrologian has a HoT stance.