Çok güzel bir açıklama, teşekkürler

If the object is in singular form (gift) that means you are talking about one thing. But the owners are many people. For that reason, I think there is no need to say as if there are many gifts. Instead, I propose using singular form of the name and 3. singular form of the suffix. Think same as ´HE´

Onların hediyesi 

OR 

In Turkish there is a form called ´Gizli Özne´ This means sometimes subjects can be hidden. You may not use the subject if it is obvious. You don´t have to repeat subject in each sentences you make. If this is what you prefer (when you need, you will use this hidden form), then you can use the plural suffıx for the verb:

.... hediyeleri ...

I understand this looks confusing but I think this language is kind of ´deductive´ or ´totalitarian´ I mean, somethings you have to memorize and then day by day the puzzle parts will become together and then you will see the whole picture. But for being fast, don´t waste too much time by asking the reasons. (my humble opinion)

As a result, if you will say with onlar for a singular noun, think same as o

onların hediyesi

onların bilgisayarı

onların evi

onların bahçesi

if you will not say ´onlar´ for singular (or also plural) noun, then use the proper suffix you already learned

(onların) hediyeleri

(onların) bilgisayarları

(onların) evleri

...

For hidden, it is still good for me ´hediyesi´ only you may want to emphasize it, and then say hediyeleri

Sen resimleri görebiliyor musun?

You have already open another thread for this:

http://www.turkishclass.com/forumTitle_56896

Maybe you didn´t realize but "plural" is "-ler" and "-lar" in Turkish.

According to Major Vowel Harmony rule,

after a - ı - o - u use -lar

after e - i - ö - ü user -ler

I think; if "gift" is not plural, you should use singular as well for "they":

bu, onların hediyesi

instead of "hediyeleri" (are there many gifts or only one gift of many people?)

Because "-ler" is here making plural the gift. Not "they".

It may look some ambiguous, but for "onlar", you shouldn´t use "-ler". I don´t know the rules in books exactly but only if the name is plural, then use plural. But if the name is not plural, I would use more singluar form, instead of making the name plural as well.

I think it is only for "going from high level into low level". As the example indicates, Bunyamin goes to Halic, most probably the seaside (low level). So it is "inmek"

düşünürüm gives an ambiguous statement. That doesn´t say any specific time. You can add that into your sentence. Otherwise it can make any meaning such as future thinking, past thinking, little or long thinking. It doesn´t say really when to think. 

düşünüyorum means thinking right now, either in the short time as continuous tense or a long time as present tense. It has time for now.

- What are you doing now?

- I am thinking: düşünüyorum.

or

- what is your opinion about this topic?

- I am thinking (for some long time), I haven´t decided yet: düşünüyorum

But düşünürüm won´t gıve this meaning. It can be anything, for example if you are talking about future conditional, you can say;

- Olursa düşünürüm; I´ll think if it happens

or if you are talking about a habit;

- Geceleri düşünürüm (I think at nights)

Maybe it means ´yarım gün´ - half day / half of the day

Yes, you can say it. As you asked the ´proper´ way, I wrote that.

-li olmak is more about the hometown / homeland, the original place of yours.

I was born in Eskişehir, so I say Eskişehirliyim. But I haven´t been living in Eskişehir for a long time.

The proper (or more popular) way to talk about where you live is ´oturmak´

So in thıs case, I am saying; Ankara´da oturuyorum. 

But if you are talking about the country more, ´yaşamak´ is the best way to use.

For example, my friend was born ın Gaziantep but he lives in Germany now. So he says: Gaziantepliyim, Almanya´da yaşıyorum.

Let´s say that he wants to tell the city he lives, he can say:

Berlin´de oturuyorum.