Do human animal hybrids exist? Are they possible via gene splicing or other technologies? Can humans and animals produce offspring?

Certainly not in the “usual” way. There are no species on Earth close enough to us to make interbreeding possible. However, there is the possibility of “genetic
engineering”, where specific genes are extracted from the DNA of another species and inserted into the genetic material of a new species (usually at the level of the egg cell).
The result can be an organism that has some characteristic of another species. For instance, they have been experimenting with inserting genes from certain spiders into the genes of sheep, and thus have sheep that produce spider-silk proteins in their milk. That’s not quite the same thing as having different species produce a viable offspring, of course.