Kyle is essentially correct. But to answer the questions:
1) PC versus Minion or Regular NC, NC wins initiative, PC tries to seize it. What happens on fail or success?
Success: PC is now the attacker.
Failure: PC is considered "undefended." From Sword Noir: "If it is a PC that is considered undefended and being attacked by NCs, the PC must make a Test against a TN of 15. The Test may only be modified by damage penalties (see Damage below). The difference between the result and 15 is used to measure the success for Damage. For example, if the result is an 8, that difference would be 7 and therefore a Good success. A result of 2 would provide a difference of 13, which would be a Great success."
2) PC versus Minion or Regular NC, PC wins initiative. can the NC seize it? Absolutely. If the NC fails, it is undefended and "The attacker still makes an attack test against the undefended character. The target number for this test is 9."
3) PC1 versus PC2, PC1 wins initiative, PC2 tries to seize the initiative. What happens on fail or success?
If PC2 succeeds, PC2 has the initiative.
If PC2 fails, PC1 can make an attack on PC2 using target number 9.
I hope that answers you question. There is also a combat example up at the website, here:
Here's the money quote, and I hope it helps:
"The first thing Tara wants to do is try to seize the initiative. Tara makes another Initiative Test but against The Cohorts’ combat Target Number. Since the Cohorts can now use their Concept of Urban Cohorts, the Target number is 14. The player rolls a 10. In the end, she’s pretty much rolling unmodified, because her Trait and Qualities are countered by the multiple opponent penalty. That 10 isn’t enough. Fail!
Unfortunately, that means the Cohorts considers Tara “Undefended.” Tara’s player has to roll 4 unmodified Tests against the Target Number 15. The player rolls a 9, a 13, a 5 and a 16.
The first roll is not only a failure, it provides Cohort 1 with a “Good” success. Tara receives 1 damage rating, she incurs a cumulative -1 modifier on all physical actions, including combat for the length of the combat, and—for the next combat turn—she may not attempt to seize initiative. The damage rating inflicts a further -2 modifier to physical actions, so a total penalty of -3. That means the next roll is a 10—an Average success, so a further -1, for a total of -4. Not pretty. That makes the 5 a 1, which is a Great success for Cohort 3—another 2 damage ratings and a total modifier of -9. That makes the last roll a 7, a Good success for Cohort 4, inflicting another damage rating on Tara, which puts her down. She receives a Doom and may have died."
Does that answer the question? Create more?