First of all, I should mention that your landlord shouldnt be able to
increase the rent for year 2001 since the new rent law limited the rents
only to 10% increase. So you dont have to increase it more than that
amount.

As for your question, you need to check out and learn if your country of
citizenship allows Turks to buy real estate in your country according to
your law system. Turkish law allows foreigners to have real estates in
Turkey only if their country allows Turks to have real estate in that
country (not only according to their legal system, but also in practical!).
If answer to that question is yes then all answers to your other questions
(mortage, share etc) are also yes.

However I have to warn you about the practical hardships you are likely
to encounter in Turkish Propriety Offices due to their lousy service even if
in theory you are allowed to do all these.

Honestly if I were you, I wouldnt bother myself to confront all these
hardships and would rather keep my rented house. My humble opinion..