What does "rely on" mean?

What does "unclaimed" mean?

In this case, "unclaimed" is an adjective used to describe things that do not have an owner - things that have not yet been claimed or possessed by someone. Thus, we can understand "unclaimed items from the lost and found" to mean items from the lost and found that have not been returned to/claimed by an owner yet. Ex: There are many unclaimed items in the subway lost and found. Ex: The unclaimed island did not have any human inhabitants.

What's "solid" mean here?

The word "solid" means dependable, reliable and trustworthy. Ex: She's very trustworthy, she has a solid reputation. Ex: John and I grew up together, our friendship remains solid till this day.

I'm curious about how to use the word "kids" in this sentence.

"Kids" in this context does not mean its literal meaning, children. The speaker has used it to imply that the others are younger, more immature, or less experienced than he is. Ex. You kids don't know how hard it was when we were young. We had to walk ten miles every day just to get to school.

What does "top tier" mean?

"Top tier" means the top level or highest level of something. In the sentence above, it means that the upper crust is the richest or most powerful people in society. Ex: Their reward was a seat in the top tier of this towering stadium, so high up that they needed binoculars. Ex: The firm services only top-tier financial advisors.

What does "we are homosexual law" mean? How can a person be law?

"Homosexual law" here is not a phrase. The full sentence here should be "we are homosexual law partners". "Law partners" refers to the fact that they are both partners in law as they are both lawyers who are partners at their job. "Homosexual" here is just thrown in for humor to emphasize that they are both also homosexual, but it does not have any meaning when attached to "law partners".