We have a comprehensive guide to Generator Essences here

To answer the questions:
What is the fastest way to get Essence to make another critter, without spending real money?
Fastest way is arguable. For specific essences, you can obtain any essence from the diamond shop (you can buy diamonds from the currency exchange for gold, if not buying from the site with money), from the Trade Broker (although that varies by what people have put there), or try making a post in the Item Auctions forum (although you'd have to specify Generator Essence at this point in time, as Frost Essences are everywhere right now)
For more info, see Section 3 of the guide

And are free-to-play players able to get their paws on the essence with Trait Points?/
Is it possible to get Essence with 6-8 Trait Points?
All Generator Essences have Trait Points. Regular Essences (blue) give 2 TP (1vis or 2 carries), while Enhanced and Mystical Essences (Purple and Yellow) give 4 (2vis, or 1vis2carry, or 4 carry)
(Section 2)

Festival Essences (Section 6) do not have Trait Points, but they're a completely different kind of item.

The closest you can get to "Essence with 6-8 Trait Points" is if you get your hands on a regennable pet (Section 7) (While it should be possible to find regennable zolnixi, no zolnixi has regennable colours, only traits, and there are no regennable bulbori, as they have yet to be revamped.)

I really want Zulnixi and Bulbori Essence... are there any tips for getting any?
Buying them or nurturing/fighting a lot of zolnixi and bulbori (finding Essences this way is a very rare drop, and may be either Regular or Enhanced.)
(Section 3 again)