My poor english seems quite a problem.
Assertion 1 just means that low level spirit will level up faster and that a spirit with the same level than the hero level up slower. That doesn't mean you should not use a spirit if it is close to character level.
Assertion 2: At each time you use a spirit in battle, next time the amount of xp decrease, and that decrease is so fast that at fourth use the xp gain is already very low.
Assertion 3: The more units there are in the battlefield the more xp the spirit will gain.
Assertion 4: Some Spirit attack does damages, for those the more damages/death they do the more xp they get.
I hope it's more clear.
EDIT: Myself I don't use those rules and don't care about spirits level up, I just focus on using them when I need and for the attack I need. I never change my attack plan for spirit xp gain, that's just me.