Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. modules over a commutative ring) it can be related to things like one-dimensionality. In the general case, I don't know whether algebras embeddable into a projective algebra may be easily characterized by their category-theoretic properties.

Are there results about this? How to characterize subobjects of projective objects?

Most likely the question whether the class of projective algebras in a variety is closed under ultraproducts is even harder, and condition might be very restrictive for all that I know. I have vague feeling that ultraproducts of projectives must be close to being just flat (although I do not know what flatness means in general varieties). I also guess this question must be thoroughly investigated. So,

Can one characterize ultraproducts of projectives in category-theoretic terms? When are they again projective? Where to read about this?

One more vague feeling - it might be that the similar question about finitely generated projectives yields more natural and more easily described answer. So, what are ultraproducts of f.g. projectives, and when are they again projective?