3 official rulings from Bandai. Straight answers to how this card actually resolves.

If after using this card's <Digiburst> effect, one of my Digimon without <Digiburst> becomes a Digimon with <Digiburst>, does that Digimon gain <Security Attack +1>?
Yes, this effect is applied as a Global Effect. If it satisfies the condition afterwards, it gains <Security Attack +1>.
If after using this card's <Digiburst> effect, one of my Digimon with <Digiburst> becomes a Digimon without <Digiburst>, does that Digimon keep the <Security Attack +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 Attack +1>.
If after using this card's <Digiburst> effect, this card leaves the battle area does the <Security Attack +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.
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Questions and answers are Bandai's official rulings, collected and translated by the DigimonCardGame Wiki community and reused here under CC BY-SA. Card names and card text are © their respective publishers; Combalo is not affiliated with Bandai.