So true, unfortunately. Why would the school charge a different price for visa conversion than they would for getting the visa at a consulate abroad? I would never pay that kind of brown envelope premium.
And then one walks back to immigration with the copy of the TM30 receipt, proud as Oscar, in order to finally get the desired extension this time, only to learn, that they would like to see the copy of the landlords house-book and ID card again (which is exactly what the landlord already presented for getting the TM30). Or something else they just made up for lack of knowing their own procedures. Oh, I could tell stories about extending at different immigration offices. Hell, I could write a book about it.
Depends on your nationality. For most, it is possible, yes. But check before about particular requirements in your 3rd country of choice. They all have different issues at times. Also, for those who can - might be easier to get an eVisa from your home / passport country (fully remotely / online).
Why bother? Visa exemption is 45 days now and can be extended for another 30 days, totaling 75 days. TR visa is a hassle and costs money and only gives you 15 days more.