Actually, not only sg 3rd+sg 3rd... 

A- speaker, B- 3rd person, C- listener D - 4th person

----------------------------------- ( kim terketmeliydi? )

...terketmeliydim diye düşünüyor ( A or B? ) 

...terketmeliydin diye düşünüyor ( C )

...terketmeliydi diye düşünüyor ( B or D? )

...terketmeliydik diye düşünüyor ( pl A or pl B? )

...terketmeliydiniz diye düşünüyor ( pl C )

...terketmeliydiler diye düşünüyor ( pl B or pl D ? )

As you see only the listener´s position is clear...

Using a name gets rid of the ambiguity:

bazen, Ahmet için uzun zaman önce terketmeliydi diye düşünüyor. (Ahmet =>4th person)

With "benim / bizim için" 1st person may be treated as 3rd person:

bazen, benim için uzun zaman önce terketmeliydi diye düşünüyor

But we don´t have to use diye...

gokuyum tried to save you from gerekmek.. but.. yes, it may have to do with gerekmek again

Welcome back to your nightmare!

Bazen, uzun zaman önce terketmem gerektiğini düşünüyor ( benim)

...terketmen gerektiğini düşünüyor ( senin)

...terketmesi gerektiğini düşünüyor ( kendisinin )

if it is 4th person, pronoun o used to make it clear:

Bazen, onun uzun zaman önce terketmesi gerektiğini düşünüyor

A name makes it even clearer:

Bazen, Ahmet´in uzun zaman önce terketmesi gerektiğini düşünüyor

...

But what is it that ties this necessity into the past? Is it only the adverbial of time? (The nature of necessity is future, it kind of belongs to its definition. This is what was bothering me in the first place.)

For sure you cannot force gerekmek into the past tense indirect question (type a in my question).

The suffix -dik ties it to the past.. You can remove "uzun zaman önce" from the sentence and it is still past necessity:

Bazen, terketmem gerektiğini düşünüyor.

Why can´t we?

"Acaba uzun zaman önce mi terketmem gerekiyordu diye düşünüyor."  

I don´t like gerekmek but unfortunately it seems to like me.

When I find a simple rule that fits my simple thinking I tend to capture it. Like what gokuyum recently wrote in another thread:

And now, scalpel, you say:

It´s not fare to compare two different statements from two teachers and two different contexts but you know my intention is pure and I am just trying to explain why it is difficult for me to understand how these sentences are organized on the time-line.

I am a big headache, I know. Actually I already got what I wanted to know. It doesn´t all have to be clear at once. There are threads which I can always dig from the bottom of the Language section pile when I find something new. The one with the headline gerekmek seems to have become one of them.

The longest (and most interesting) discussions are often about the minimalism and ambiguity of Turkish sentences. Turkish expressions are tied to their context in a way which is different from any other language I have learned. Some philosophy to the end:

-dık (-dik, -duk, -dük, -tık, tik, -tuk, -tük )  is one of the most widely used participle suffix (verbal noun) indicating past. One of its noticeable features is that it is almost always used with personal endings. It indicates past but it is not a tense suffix. We  always know the "tense" from the main verb (predicate)..

ne yaptığımı gördü (past)

ne yaptığımı görüyor (present)

ne yaptığımı görecek  (future)

Let´s rewrite the sentences (to help you to see time-line thing) with another participle -acak, -ecek which indicates future:

ne yapacağımı gördü (past)

ne yapacağımı görüyor (present)

ne yapacağımı görecek (future)

Do you think, for example, ne yaptığımı gördü and ne yapacağımı gördü could be of the same meaning just because their predicates both are past tense? 

I don´t think anything, scalpel. I am just trying to imitate native users, often with plenty of work and little success. What I am asking here all the time is models to follow.

I wonder if we could put it this way: the tense of the predicate tells us the point in time from which we look at the action. What is it called? Reference point? Then we choose which one of the personal participles to use:  -ecek- or -dik-. The division of labour between them, could it be

         -ecek-: future, the time after the reference point

         -dik-: everything else ?

You are already halfway up.. you will reach the peak soon.. yes, "the tense of the predicate tells us the point in time from which we look at the action." .. but the above mentioned participles are always in harmony with the tense of the predicate..

Gideceğimi biliyordu - he knew I would go

Gideceğimi biliyor - he knows I will go 

As you see in the examples above gideceğimi changes its meaning for the sake of the harmony with the tense of the predicate, but never lose its fuction indicating future..

-dik indicates past.. a thing that happened before the reference point..

Gittiğimi biliyordu - he knew I had gone

Gittiğimi biliyor - he knows I went

What confuses you is the examples like , "Onu sevdiğimi biliyor".. -dik suffix behaves the actions all the same, and gives no privilege to continuing actions.. they may be continuing but they are grammatically finished..

What about the action which is simultaneous with the predicate verb, which takes place at the time of the reference point? Isn´t it -dik- also? That´s what I suggested: -dik- covers everything else but future.

This is not the strange thing. What is odd is the Turkish use of present continuous with verbs denoting feelings. In my opinion it would be more convincing in these utterances to use the aorist which expresses this action is done at all times and it somehow belongs to the speaker´s habitus and nature?

-dik- suffix is used for past and present but it is not used for future, for future, you should use -ecek-. But if it is important to indicate the time, you can use the structures below:

example sentence: Yaptığımı biliyorsun. (probably, this would be understood as "You know that I do." by the listener.)

• for present continuous: Yapmakta olduğumu biliyorsun. or Yapıyor olduğumu biliyorsun. ("You know that I am doing.")
  
• for past: Yapmış olduğumu biliyorsun. ("You know that I did." or "You know that I have done.")

ne yaptığımı gördü and ne yapacağımı gördü could seem to be of the same meaning but ne yapacağımı gördü is actually ne yapacak olduğumu gördü and this sentence means "he saw what I was going to do" and ne yaptığımı gördü means "he saw what I did/was doing". So they are not in the same meaning actually, there is a slight difference. Also other sentences are not in the same meaning, too. Ne yaptığımı görecek: He will see what I do. Ne yapacağımı görecek: He will see what I will do.

I would like to share something about gerek. In question sentences you can use a difference structure with gerek.

For example:

• Telefonumu almam gerekiyor.
• When you will write this sentence as a question, you will probably write:Telefonumu almam gerekiyor mu?, which is the normal question structure.
• But you can also use this: Telefonumu almama gerek var mı?, this is a very common structure in Turkish.

Another example:

• Gitmem gerek. (I must go.)
• Gitmeme gerek var mı? (Must I go?), to which you can answer as "evet, gerek var" or "hayır, gerek yok".