If after using this card's ＜Digi-Burst＞ effect, one of my Digimon without ＜Digi-Burst＞ becomes a Digimon with ＜Digi-Burst＞, does that Digimon gain ＜Security A. +1＞?
Yes, this effect is applied as a Global Effect. If it satisfies the condition afterwards, it gains ＜Security A. +1＞.
If after using this card's ＜Digi-Burst＞ effect, one of my Digimon with ＜Digi-Burst＞ becomes a Digimon without ＜Digi-Burst＞, does that Digimon keep the ＜Security A. +1＞?
No, this effect is applied as a Global Effect. So if it no longer satisfies the condition then it no longer has the ＜Security A. +1＞.
If after using this card's ＜Digi-Burst＞ effect, this card leaves the battle area does the ＜Security A. +1＞ continue being applied?
Yes, as it has already resolved and applied a Global effect, the Global effect continues applying even after Rosemon leaves the battle area.
