Does homology of a trait necessarily imply that the most recent common ancestor had the trait?
Yes, assuming you treat the term strictly.

The term homologous implies that the most recent common ancestor had the trait, although it would be possible for the term to be used in error, when something thought to be a homology is actually not.

If the trait is shared but the trait evolved separately twice, instead you would refer to it as an analogous trait rather than homologous; analogous traits are often the result of convergent evolution.

I can imagine some edge cases, however, where one could argue about homology, although I don't have a specific example in mind (maybe someone can help me out if they can think of one). Imagine you have a trait that is present in an ancestral population, becomes minimized in a common ancestor, and then multiple species descending from that ancestor start to exhibit the trait again. This is called atavism. In some ways, atavism could be seen as convergent evolution and an analogous trait: each has separately evolved that trait from the last common ancestor. However, one would expect a lot more genetic similarity than other examples of convergent evolution and therefore outside of cladistics it might be more appropriate to think of it as a homologous trait in that you can find some ancestor that shared the trait, it just doesn't happen to be the most recent one.