Can I carry on using the roses and getting the titles, new quests and achievements once the event finishes - if I gather 2000 roses now?
Absolutely! You'll often find folks flinging them around every time there's another event that causes crowds to gather.
I don't know if this is how it always used to be. I remember having to stockpile on roses one year in hopes of giving out all 2000 the next year during the E-day event because the progress was reset every year... or at least it seemed that way. Fortunately, I was diligent and gained "The Prurient" title back around 2009 or 10.
Since the Achievements are keeping track of how many roses you've given out, the number doesn't reset. It's like total number of killed things.
It only started the count for the next achievement in the line at zero once, I believe. It happens when more achievements are added to an achievement once you've already earned earlier achievements. It is a known issue with the achievement system.