“My X” tends to imply that there is only one X. This can mean that you only have one X, or that it is clear from the context which X you are talking about. This is not absolute: depending on what X is and on the context, it may be strongly implied, or weakly implied, or even sometimes not implied at all.
For example, “this is my friend Lord Henry” sounds perfectly natural, because (presumably) you only have one friend who is called Lord Henry. On the other hand, “this is my old friend” sounds a little weird: it sounds like you have a single old friend (meaning, a single elderly friend). In this specific case, there's an additional issue with the adjective “old”. In “this is my old friend”, it could either mean “elderly friend” or “person who has been my friend for a long time”. On the other hand, “an old friend of mine” can only mean “a person who has been my friend for a long time”.
To my non-native (but fluent) ears, “this is X, a friend of mine” and “this is my friend X” and “this is a friend of mine” all sound natural. “This is my friend” only sounds natural if you've just been talking about that person, and it then means “this is the friend that we were talking about”.
Nonetheless, there are contexts where it's perfectly idiomatic to say “my friend” without implying that you have a single friend. For example, in the sentence “he's my friend, I won't let him down”, there is absolutely no implication that you have a single friend. I find this particular more idiomatic than “he's a friend of mine, I won't let him down”. I don't know why; maybe it's because “he's my friend” makes your connection closer and this is an important impression given the sentiment expressed by the sentence?
Similarly “a teacher of mine said …” is the natural way of saying that one of your teachers said something without specifying which teacher. “My teacher said …” either refers to the specific teacher that you have been discussing earlier (it's equivalent to “the teacher [of mine] we were discussing earlier said …”), or implies that you have a single teacher.
“I've met many friends of yours” could equally be “I've met many of your friends”. The first sentence takes “a friend of yours”, pluralizes it to “friends of yours”, and adds the determiner “many” to indicate that the number is more than a few. The second sentence takes “your friends”, and adds “many of” to indicate that the number is more than a few but less than all. The meaning of the result is the same.
“This is a house of ours” implies that you have several houses. “This is our house” implies that (like most people) you have a single house.
“I've discovered much useful of yours.” is not grammatical: it's missing a noun or pronoun at the core of the direct complement of “discovered”. I don't understand what you mean to say.