Yes.

[Which might seem to be the least helpful answer ever, until one realized that it was technically true. Regarding each question Anya asked, albeit in different ways:]

I am generating it. It requires constant maintenance, but is currently stable due to the near constant source of energy contained within my person--one not even a Barge Coma was apparently able to neutralize. Only upon my "death" would the energy fade.

And, yes, there are different forms of constructs.