It's legal to say something like, "When did you go to the cinema last?" although I prefer it as in your first example, "When did you last go to the cinema?" Adding "time" to the end of that is not natural English, though.
There are, of course, many ways to end a sentence with "last time", as long as it makes sense in context and (I think) is part of a phrase:
Can we go to the cinema like last time?
Which cinema did you go to last time?
Did you like the movie we saw last time?
Who were you with last time?
And so on. The difference between these and your question is that "last time" is the direct object of the question, "when was the last time ..." and the rest is an adjectival phrase that modifies "time". With the other examples, "last time" is part of a phrase that modifies some other part of the question.
too / either indicates that one person or things is (to be) treated the same as one that has already been mentioned. too is used in positive sentences:
I have told Jane. - positive
I have told Mary too. - positive
either is used in negative sentences, but only if the first sentence is also negative:
I haven't told Jane. - negative
Don't tell Mary either! - negative
If you have a positive sentence followed by a negative sentence, you use too, even though the second sentence is negative...
I have told Jane. - positive
Don't tell Mary too! - negative
Adding forget to doesn't really affect the choice of adverb:
I have told Jane. - positive
Don't forget to tell Mary too! - negative
So, in your sentence, the two parts are:
I'll write her a card. - positive
Don't forget to write one, too - negative
too is the correct adverb to use, even though the second sentence is negative, because the first sentence is positive.
Note that, if both sentences are negative, you use either:
I won't forget to write her a card. - negative
Don't forget to write one, either - negative
Best Answer
'Social media' has become a generally acceptable and widespread understood term, so you are fine to use this without looking like a dummy.
While 'social network sites' has a more formal tone, recently I have been seeing the term 'social media' also more and more in formal reports, so also in this setting I would deem that acceptable.