Heleni1990

You can inform your school.  They will update your I-20 accordingly.
 
Your student visa status/history has no bearing on your K1, unless you overstayed in America after your F1 expired.  This is not your situation so there is no impact.
 
It is perfectly fine for your fiance to visit you in Greece during the process.
 
In your case, if you two truly want to get married, the  marry now and file for the CR1 spouse visa.  This will allow you to be an LPR as soon as you enter the US and start work and drive right away, and travel back to Greece to visit family.  
 
With a K1, you will need to wait 6 months to 2 years to get employment authorization, advanced parole and adjustment approval.

I don't see any issues.  I am not completely familiar with student visas, but if you did not overstay in the US, it should have no bearing on a future K1 visa process.  As to my knowledge, you can drop school from anywhere, and I am not aware of any requirements for you to go back to the US on your student visa.
 
One thing you should do is thoroughly research the difference between a K1 visa and a spousal visa (CR1/IR1).  There are great limitations and downsides of a K1 visa especially after entering the US and getting married.
 
Good Luck!

Sorry. I should have written no

It is slower.
 
Because you will enter rhe U.S. as an LPR able to work and able to travel internationally. As a K-1 you will wait 2 more years for a gc, and sit around the house bored. 
 
Not in the experience of thousands of visa journey members.

Yes
 
After filing you continue to meet, and do what is possible to unify finances. So for example, add each other as authorized users to your credit cards.

Obviously most people in your situation are in the same situation.