Tamamo is wrong.

@Ebenheizer then you said it wrong (if its about your lineup). Numbering of collumns and rows always starts from left to right (collumn) and from left to right (row). It's the same about opponents, but you can read about that below.

When you enter ninja tab, you have a grid that looks like this:

x x x

x x x

x x x

General order of positions in this grid is:

7 4 1

8 5 2

9 6 3

and for your opponents during a fight (during a fight, not when you inspect their team) is mirror version of this, so whole battlefield looks like this:

7 4 1 | 1 4 7

8 5 2 | 2 5 8

9 6 3 | 3 6 9

Order of your ninjas movement depends where they are placed. Lower number on this list moves before the higher number, same goes for your opponents. For example, you have ninja on grid 3, 5, 8 and 9 (underlined)- and this is the order in which they will go. Ninja on grid 3 is on position (or move) 1, ninja on grid 5 is move2, ninja on grid 8 is move3 and ninja on grid 9 is move4. Opponent might have ninjas on grid 1, 2, 3, 4. In this case initiative is compared between positions, so your ninja on grid 3 vs opponent ninja on grid 1; your ninja on grid 5 vs opponent ninja on grid 2, etc.. There's no difference if your move1 is on grid 1, 3 or even 6 (except for tactics pages bonus stats) in terms of how initiative works, as long as there are no ninjas that are placed before him (lower numbers of grid above). If you are still unsure if your ninja is on what move, when you enter ninja tab you have big red numbers with number of action.

It is possible for a ninja on a higher grid number (lets compare setup from example above and ninjas on grid 3 and 5) to take action before lower grid number (in simple words mvoe2 goes off before move1). For this to happen, ninja on move 2 has to have 10k initiative more than your move1. If your grid3 ninja has 8000 initiative and your grid 5 ninja has 19000 initiative, grid 5 will take action before grid 3. Same applies to other positions between them. It also applies to you vs your opponent. Quick example:

Initiative of your ninjas...... opponent ninjas

move1- 18k..........13k

move2- 25k...........12k

move3- 10k.......... 9k

move4- 10k............11k

Order of movement in this scenario is as follows: 1st. your move1, 2nd your move2, 3rd opponent move1, 4th opponent move2, 5th your move3, 6th opponent move3, 7th opponent move4, 8th your move4.

Interesting fact: its possible for a ninja on move4 to act faster than move1. For this to happen, position 4 has to have over 30k ini advantage over your move1.

@Ebenheizer

"Let me make a scenario here:

Same with your formation example, say, the pos 1 of my opponent is 20K. 2K higher than my pos 1 ninja, but 5K lower than my pos 2 ninja, then will the opponent (with lower initiative) goes first instead of my second pos (with higher initiative)?"

Yes, in this case you mentioned your opponent's move1 (20k) will go first. Then acts your move1(18k) and your move2 (25k). For a moveX+1 to go before moveX the difference in initiative needs to be over 10k. In this case the difference between your move2 and move1 is 7k, and between your move2 and opponent move2 is 5k, so move2 cant go faster. Scarlettblue might have said it in easier words in his comment.

Regarding another question:

Yes, your ninja will move first. It doesn't matter if opponent is in the frontline and you arent.