There is something I can´t get about the -DIk participles.

When we use it to make one sentence out of two sentences the object may be in any case:

• Köpeği (acc.) gezdirdim. O uyuyor. -> Gezdirdiğim köpek uyuyor.
• Bir köye (dat.) gittim. Orası güzel. -> Gittiğim köy güzel.
• Rüzgârdan (abl.) ürktüm. O tuhaf tuhaf uluyordu. -> Ürktüğüm rüzgâr tuhaf tuhaf uluyordu.
• Bu evde (loc.) yaşıyorum. O üç katlı. -> Yaşadığım ev üç katlı.
• Komşumla (ins.) konuştum. O gelecek. -> Konuştuğum komşum gelecek.

However, what bothers me is that there is no trace of the original case endings in the resulting sentences. The information carried by the case endings seems to be lost. Thus, "the village I came to" and "the village I came from" both seem to be translated as "geldiğim köy", which doesn´t make sense to me as those are kind of opposite things. What am I missing?

You are not missing anything. That´s one drawback to attaching the prepositions to the end of words, you can´t use them separately. It might seem illogical to an English or German speaker since the prepositions are used very often in those languages, but usually the meaning is inferred from the context although there are cases where ambiguity occurs as in your last example. If you want to be absolutely precise, a not so natural way to remove this ambiguity is to add words with the appropriate prepositions to the sentence like:

İçinden geldiğim şehir : The city inside which I come from

İçinde yaşadığım ev : The house inside which I live

Huge thanks for making it clear, gugukkusu. I couldn´t find this information in any grammar book I checked. I´ve got one more question. What about the postpositions that are separate words? For example, karşı. How do I say "the people we´re fighting against" in Turkish?

Well, again you would say "Savaştığımız insanlar" because its not actually used separately, the correct form is "-e karşı". You can, however, use certain words if they include a preposition in themselves. Some examples off the top of my head:

Hakkında: Hakkında konuştuğum konu The subject I´m talking about

Aleyhinde: Aleyhinde konuştuğum kişi The person that I´m talking against

Adına: Adına mutlu olduğum kişi The person that I´m happy for

Uğruna: Uğruna savaştığım şey The thing that I´m fighting for

 Üstünde, içinde, dışında: Üstünde (içinde,dışında) olduğum yer. The place that I´m on (in, out of)

Birlikte: Birlikte gittiğimiz yer The place that we went to together

And what happens with "için"? Does it disappear?

"The child I bought the present for is my nephew."

=

"Hediyeyi aldığım çocuk yeğenim." ?

Yep, usually it does. Sometimes you can use other words instead though, as you can see in my examples with "for".

In "Turkish: a Comprehensive Grammar" by Aslı Göksel and Celia Kerslake, I found that kendisi/kendileri can also be used to carry the lost case suffixes and postpositions if the object is a human being.

So it seems to be possible to translate "the people we are fighting against" as "kendilerine karşı savaştığımız insanlar" and "the child I bought the present for" as "kendisi için hediye aldığım çocuk". Right?

How often is this structure used? How natural does it sound?

Are these kendisi/kendileri here just to carry the case/postposition, or do they add some additional meaning?