I was trying to understand these:

Sevdiğim - (-dik) Personal Participle

Seveyim - Subjunctive Mood

Are these common tenses in Turkish? What do they mean?

-dik is not a personal participle. It makes verbs adjectives. And that  "im" is possesive suffix there. Ex: Okuduğum kitap: The book I read. 

You can use it as a noun too. Ex: Sevdiğim geldi. My beloved came.

Seveyim means let me love.  sev-e-y-im  "im" is personal participle here.

Here is some information about subjunctive mood: http://www.turkishlanguage.co.uk/submood.htm

Thanks. Makes sense with beloved since in the rest of the sentence there was a verb so I was confuses why there would be 2 verbs, so now your explanation, it is changed from a verb.

Can I say "Bu pisirdik tavuk" for this is cooked chicken?

I googled Personal Participle, some sites insisted it was "dik" but then again I could not find this term in English grammar. Doesn´t matter, I just wanted to understand the meaning.

You cant use dik with pişirmek. There are other suffixes like -dik which make verbs adjectives. You should use the proper suffix for that verb.

Here are the suffixes:

-an, -ası, -mez, -ar, -dik, -ecek, -miş

We use suffix miş with pişmek or pişirmek.

"Pişmiş tavuk" means cooked chicken.