It is assumed that similar sequences perform similar functions. Is the converse also true?
Fairly similar sequences are likely to perform similar functions. Let's call those similar proteins members of protein families.
Somewhat similar sequences will perform somewhat similar functions, and those members would belong to protein superfamilies. A while ago I described a superfamily of phosphoesterases that are very different in terms of sequence, yet they all cleave phosphate esters. When I tell you that some of them are phosphatases and others are nucleases, you may think they are nothing alike. Yet both single-phosphate removal from any substrate and a breakdown of a phosphodiester backbone in DNA/RNA are chemically identical reactions, so these enzymes perform
somewhat similar functions even though their sequences often share less than 20% identity. There is a whole spectrum of what
similar means, and the meaning will be different when applied to the chemical nature of reactions vs chemical properties of substrate. So the answer to your second question is no.
Your first question is unclear, so I will only generalize by saying that functions tend to be preserved better than sequences. There are relatively few residues in almost any enzyme that have to be absolutely conserved, and that is usually because the reaction chemistry requires particular amino-acid side-chains. There are usually several other residues in or near the active site that are involved in substrate recognition, and they are typically well-conserved but not as much necessarily as catalytic residues. Almost everything else is flexible to a good degree.
Let's say that each human has 5 catalytic residues at the tips of our fingers. If I give you a baseball that has 5 dots on it, almost any person in the world would be able to line up the tips of their fingers to those 5 dots, regardless of how big or small their hands are. This wouldn't be the case if I gave you a ping-pong ball or a basketball, as people with large hands could not handle the former, and those with small hands the latter. The point is that as long as a small number of catalytic residues are preserved and they are spatially organized in a defined way, the exact scaffold (small hands, large hands, young, old, men, women) that brings them to that spatial position is not very important. That's why enzymes can retain the same (or
similar) functionality even when their sequences change to a good degree.