Why "på" and not "om"? Is it special to the verb "att tänka"? Could "Jag tänker om dig" be correct?
"Vad ska hon tänka om mig?" is possible but as far as I know "tänka om" is only used together with some specific though or question about a thought. "Hon tänker gott/illa om dig."
Is there a reason you use tänka instead of tänker I seem to see a lot of words shortened this way or by dropping a letter. Is it simply a shorter spelling or has the meaning of the word changed at all?
Difference between this and tycker? I've seen tycker used colloquially in the context of "if you think I am a good friend"
It means "think" on its own. So you could use it in phrases like "Hon sitter och tänker" - She is (sitting and) thinking.
But as soon as you want to add an object you need the preposition.
Read the comments but still the Bogan is not sure. So I ask why not, "I think of you?"
That works too (and it's an accepted answer). The Swedish sentence covers both.
How can the difference between "thinking of something" and "thinking about something" be made clear?
That seems to be hard. For instance this dictionary gives them both the same definition: http://dictionary.cambridge.org/dictionary/english/think-of-about-sb-sth
On a more serious note, I understand that you meant 'in Swedish'. But since the difference isn't clear in English, the answer is that we translate depending on what is meant. So if you can replace 'think of' with 'mean', you can use mena. If you want to say something never crossed your mind, as opposed to not having thought about it seriously, you can say det har aldrig fallit mig in ('it's never occurred to me'). If you want to make it clear that you've pondered something as opposed to it just crossing your mind, you can use the verb fundera (på) instead. And so on, I suppose – there are probably more possibilities that didn't cross my mind now that I thought just briefly about this :)
It may be considered a bit late to ask this but when is "ä" pronounced as "e" like in "verb" and when is it pronounced like as "a" like in "cat" ?