Yes, you should be able to synthesize or produce any sound imaginable with just the built-in instrument and effects plugins. It would be worth it to review some tutorials on basic sound design then apply those ideas to Impact, Mai Tai, Mojito, Presence, and Sample One.
The easiest path to non-instrument sounds would be downloading one of the many free sound effects and/or folley sample packs and then just loading those samples into Sample One or Impact.
The next step would be to synthesize non-instrument sounds using Mai Tai, Mojito, or Presence. Sirens are pretty easy. LFO modulate a pair of oscillators. Fireworks you would have to think through. For instance, synthesizing the sound of a bottle rocket would probably take more than one instrument. You would synthesize the whistle as it launched, and the pop as it explodes.
There are patch books on the internet for older synthesizers like the Arp 2600 and the Roland SH-101 which include patch instructions for all kinds of non-instrument sounds like helicopters, lightning and thunder, dogs barking, etc. You could learn from those patch instructions and apply the basic techniques to Mai Tai rather easily.
The new Korg Arp 2600 patch book has patches for both Sirens (page 69) and explosions (page 78). Sure the Arp 2600 is laid out differently than Main Tai, but it's the same basic idea: oscillators, filters, envelopes, vca, lfo, noise, etc.