It only means that the sequences are similar enough that we can call them different versions of "the same" sequence (in the same sense as a human arm and a bird's wing are different versions of "the same" body part). Sequence homology doesn't necessarily imply common ancestry; as with other traits, it's the pattern of similarities and differences that agrees too well with evolutionary predictions to be an accident.
Sequences that can't be aligned can result from all sorts of things. Different sequences evolve at different rates, and if a sequence is "fast" enough or the organisms diverged long enough ago, no detectable similarity remains between them. Fairly large chunks can also be inserted (for example, by a virus) or deleted in one of the organisms, and of course you won't find similarity between something and nothing.
Do you mean aligning sponge sequences to human sequences, or sequences from animals between humans and sponges to human sequences? (A "genome" is all genetic material an organism has, BTW)
In either case, the answer is yes (there's a reason they say the human and chimp genomes are over 90% similar!).
Yes. I'm a bit lazy to look up studies, but giving you an example takes a bit less work. Let's look at
cytochrome c, a small protein found across the tree of life. I took
this human cyt c protein sequence and searched (using
BLASTp) for similar proteins in bacteria.
The search turns up a number of bacterial cyt c proteins that are up to 51% identical and up to 70% chemically similar to the human protein.
(To put this into perspective: since living things use 20 amino acids to build proteins, two random protein sequences of the same length are expected to be about 5% (1/20) identical)