Do note that Refining can be a waste as well. Going first isn't always the optimal play and Water mains tend to play reactive style. Kakashi is in position 3 so anyone like Neji or Midnight Blade who are usually in position 1 or 2 will go before he does even if you boost your refining. Refining can be important but there are ways to get your jutsu in with good timing instead of refining stats. That said, crit chance and combos are also refining dependent so up to you how many coupons you want to waste on RNG.

Mood however is a huge help that is guaranteed. Your water ninja with a few mood scrolls can boost her ninjutsu by 30% and be healing like crazy and nuking things left and right. Or if Kakashi constantly dies then consider boosting his Resistance or Life by 30% to make him more durable. Moods go beyond 30% but that's an affordable goal.

"So lets say my Kakashi is in position three, and an enemy Kakashi is in position two, and they both use their mystery skills. Are you saying that the enemy Kakashi will always pull off his mystery skill first regardless of initiative because he is in position two and my Kakashi is in position three? "

Yes, he will always pull off his Jutsu first. Your position supercedes Initiative. That's why interrupts like Iruka and Neji being in the front tend to be awesome while Gai and Shikamaru being in the back tend to be lousy.

Initiative is just a tiebreaker. The game checks the following to determine who goes first...

Priority 1 - Type of Attack - Prompt Jutsu always go before Standard Jutsu
Priority 2 - Position Rank - Ninjas act in number order, move your ninjas up to go faster
Priority 3 - Initiative Stat - The last thing that matters and a tiebreaker for same positions

I'm sure there's something that determines how you act if your initiative is identical. Just don't know what it is.

Also note that in PVE Initiative matters more. The enemy won't act if your initiative beats theirs which is why your entire team will go before them sometimes.