It's just worth noting that there exists a prescriptive etiquette to avoid the construction se rappeler de.... Of course, it is just a piece of etiquette, and it's certainly not one that's generally followed by many speakers in everyday speech. But I'm guessing that speakers that do buy into this etiquette might instead use Vous vous souvenez de moi? (which interestingly also appears to get a much higher frequency count on Google).
Almost! Just like in English where you say "I remember" but "he remembers", in French, you also need to make the subject match the verb. So it turns out to be:
Je me souviens de toi.or Je me rappelle de toi.
There are two versions because French has two verbs meaning 'to remember'. Note what I mentioned that the second of these, although common in informal speech, would be avoided by a few speakers.
To say "I remember meeting you...", you san say, for example:
Je me souviens d'avoir fait votre connaissance l'an dernier/en 2005
More informally, you could say something like:
Je me souviens/rappelle qu'on s'est rencontrés en 2005
OK, there are kind of two answers to this, depending on whether you're interested in "the reason" in terms of how the language works today, or in terms of historical derivation.
In terms of how the language works today, it turns out that what we informally call "reflexive" verbs aren't actually always reflexive-- in the sense that the extra pronoun (me, te etc) doesn't really have a reflexive meaning. In other words, it doesn't really mean myself, yourself etc in any literal way-- it's just there "because it is". Other examples include verbs such as se moquer de..., se rire de... meaning "to mock...". We would probably want to say that se souvenir de... falls into the same category.
The verb se rappeler (de)... is a little bit different. In this case, there is more of a sense of it literally meaning to remind oneself of.... Clearly, the two verbs have influenced each other as they've evolved.
(For native speakers of French: if you don't get the difference between, say, se laver and se souvenir, consider what happens if you use ne ... que... to say e.g. "She washed only herself"; you'll see that this construction doesn't work with verbs like se souvenir.)
Now, as far as the historical question is concerned, what appears to have happened with se souvenir de... is that originally, the verb souvenir (Latin subvenire) meant something closer to "to come up to...", "to come to the mind of...". So originally, people would have said something like (Il) me souvient de... to mean Thoughts come to my head about...-- i.e. "I remember...". This construction is no longer everyday French, although in literary usage, you can still just about use Il me souvient que... to mean "I remember that...".
Now, there's a fairly common phenomenon in languages whereby the subject/object of verbs can become transposed. This appears to happen because speakers don't fully analyse sentences: they just use common sense to work out which is the subject and object (e.g. when presented with sentences like "The teacher was graded by the student", speakers actually sometimes interpret that as meaning "The teacher graded the student", because without fully analysing the sentence, that's what common sense tells you). So over time, (Il) me souvient de... became re-analysed as though me reflected the subject of the sentence, which then became je.
"Je vous ai rencontré il y a quelques temps" is correct. But it's more usual (and it sounds less "selfish") to use "Nous nous sommes rencontrés il y a quelques temps" (written) or "On s'est rencontré il y a quelques temps" (spoken), ie. "we met each other" instead of "I met you".
Now, "do you remember me?" would be "Vous souvenez-vous de moi ?" (written) or "Vous vous souvenez de moi ?" (spoken).
"Do" can be translated to "faire" only when it's the main verb, not as a word to create a question (here, the real verb is "remember" - "se souvenir").
2. sometime
I will take vacation sometime in July and go to France.
[ I don't know the exact date so I use the word sometime.] ( I will go to France. No doubt about it.)
3. some time
I am making lunch now. It will take some time to cook food.