You omitted the most important fact ... your 
Lyanna Natura is currently 
Level 7 so far! That alone would give you an advantage over me, were we to battle together. It means that you can unlock the higher level Abilities of your Monsters. I was collecting Summoners hard out, long before the announcement of how valuable they would be in battle was widely understood. Unfortunately, once that announcement was made, the price of the Summoners skyrocketed, and I can't really afford them much these days.
All the best for your Collection and the battles ahead. I have no idea whether two players using the same Splinter can battle against each other, but I'd presume so, judging by everything I've read so far about the game.
Yes that is true my Lyanna is Lvl. 7 right now and it is expensive to get there. It would be great if the two same splinters can fight in PvP against each other so you could figure out what are the best Monsters to choose and which positon suits them best. Good communication with you lets stay in contact!