Yes, I understood, thank you so much! 

Maybe that is because I am a great teacher

Yes! 

You know how to spoil me

There is some more explanation on when to use buffer letter -y here: 

http://www.turkishclass.com/forumTitle_62062

About the use of the buffer ´y´:

http://www.turkishclass.com/forumTitle_61671

1. past to be + adjective: yaşlıydı, he was old

2. past to be + noun: siyah bir arabaydı, it was a black car

3. And furthermore all past* suffixed verbs ending with a consonant. 

- past conditional: olsaydı, if it was

- past necessitive: aramalıydım, I had to call

- past subjunctive: tıklayaydım

*Past being either -di or -miş

Follwing that rule: başla+mış+ım is without y because the verb stem is not suffixed with -sa or -malı or -yayım or such.

And it is başla+sa+y+mış+ım because başla is suffixed with the conditional suffix -sa.

Why we don´t add to anladım? Because anla isn´t suffixed or?

What heppend with (possessive) pronouns? Benim/Ben, how to say It was mine/me?

Anla is a verb. If it was a noun you must add the buffer letter.

anlasaydı(if he had understood)now I added a buffer letter because there are two moods(first one -sa, second one -dı) and -sa ends with "a".

It was mine: Benimdi.

Thank you.