Yes mood stays between switches, strengthening equipment should be prioritized over group skills. (this due to the convoys giving you enough for grinding gs as needed.)

And of the 8 inner gates, it's best to worry about Nin,Res,def, and crit early on, then combo,injury, regen, and hp later on as needed.

Re-mooding each main switch isn't really too important unless you pvp.

---ofc this is just my opinion--- be you, see what works.