You people, I swear..
His whole effect after "When this Ninja is put in play" is a triggered effect that applies to your field. It triggers, everyone gets +2/+2 and if a Team wins an OV, you get a BR.
You can then Flashpoint him and the effect Will stay.
The free BR effect applies to any number of times they win an OV, so three Teams getting OVs is 3 BRs. Battles are processed separately, and his effect doesn't say "You can win 1 BR," So it applies to every instance.
Zac is completely wrong on all accounts, and once Tylar or Andy comes in and does nothing else except quote my post saying "Shino'sDad is right", they'll even mean "He's right that Zac has no idea what he's talking about."