The mek infinitive is often found with ´icin´ to give the meaning similar to ´in order to´

for example - tenis macini seyretmek icin eve gidiyorum

I´m going home to see the match (in order that I can see the match)

Turkiye´ye gelmek icin biletimi yeni aldim - I just brought my ticket to come to Turkey

Oraya gelmek icin ucuslara bakiyordum - I was looking at flights (in order) to come there

Ah Thank you Inşallah! Makes sense to me now

Actually I need to study this more. Still a bit confused as to when to use this. What you are describing here is doing one action to facilitate another action, or as a means to do the next action? 

Can the verb be in passive or other tenses and still take -için after, so long as the action is still infinitive? Could one use açtırılmak the same way? Or is it that if the tense of the verb changes, then it can´t be followed by -için? would the following sentence convey this correctly?

kapıyı açtırılmak için, anahtaryı birinci kullanmalısınız. For the door to be opened, you must first use the key.

And another practice sentence:

Türkiye´ye gelmek için, bir pasaport taşımalısiniz. -In order to come to Turkiye, you must carry a passport.

 açtırmak is causative form of the verb açmak.---> to make something opened.

 Kapıyı açtırmak için bir çilingir bulmalısın. --> You must find a locksmith to get the door opened.

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 açılmak is passive form. ---> to be opened.

 Kapının açılması için biraz bekle. --> Wait a while for the door to be opened.

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açtırılmak is more complicated form that consists of causative and passive suffixes.

Müdür kapının açtırılması için emir verdi. ---> The manager ordered to get the office door to be opened. [ The manager ordered people to find [get]someone to get the door opened ]

So için can come after causative form of a verb (açtırmak), passive (açılmasıand passive/causative (açtırılmak), the latter two needing -ma +plus the -si of the possesed object in an izafet relationship? (kapının açılması)

What is the case -me/-ma (if not  for negation) + si? I was also told that verb combination was needed for the following sentence:

Ankara´ya gitmesi gerekecek mi? -Will he need to go to Ankara?

Which does not have the possesor suffix after Ankara?

aç|ıl|ma|sı = stem + passive marking + infinitive marking + poss sg 3rd

git|me|si = stem + infinitive marking + poss sg 3rd

You see, Mavili, -me- here marks the infinitive. And you are right, confusingly it looks the same as negation (it´s good you noticed so that you can keep an eye on it in the future).

-me- changes the verb into a noun. And once it is a noun you can paste possessive suffixes and other material into it as you like.