Well, your approach was very logical. You almost got it right. It is like this:

"alabilir miyim"
al / abil / ir / mi / y / im

1. 'al' is the verb stem of 'almak'.
2. 'abil' (or 'ebil') is the abilitative suffix. It has an etymological connection with bilmek as you guessed.
3. 'ir' is simple present tense. Instead simple present (which is the most common with ebil) you could use another tense: "gelebildi mi?", "gelebiliyor mu?", "gelebilmiş mi?"
4. mi is the question suffix.
5. -y is buffer letter that comes inbetween two vowels.
6. -im is personal suffix taken from this list:

Singular
1. : -ım, -im, -um, -üm
2. : -sın, -sin, -sun, -sün
3. : --
Plural
1. : -ız, -iz, -uz, -üz
2. : -sınız, -siniz, -sunuz, -sünüz
3. : -lar, -ler

Oh, I see. I thought the the lady asks people whether or not they want tea. If people ask the tea lady than this is what they say:

"Çay verir misin?"
"Can you give me a tea."

'Verir' is a conjugation or 'vermek' (to give).

This is not actually a very polite way of asking for something. It sounds a little patronising to me.
A more polite version would be :

"Bir çay alabilir miyim lütfen?"
"Can I have a tea please."

Hi,

"çay ver misin?" doesn't exist.

It can be one of those:

"Çay içer misin?" Would you like to drink a tea?
"Çay ister misin?" Would you like to have a tea?

'içer' is the simple present conjugation of içmek.
'ister' is the simple present conjugation of iistemek.

It is more polite to speak in plurals. These are preferable:

"Çay içer misiniz?"
"Çay ister misiniz?"

These are common answers in this case:

"Evet, lütfen."
"Yes, please."

"Hayır, teşekkür ederim."
"No, thank you."

Greetings,
You can see all possible conjugations with winmekmak. Simply choose -meli as tense and choose a modifier. Here is a screenshot:
http://img408.imageshack.us/img408/9535/winmekmakfc5.jpg

Here is winmekmak listed on #21 on my list:
http://www.turkishclass.com/forumTitle_6_6212

Cheers,
Erdinç

As others told there are two versions for university graduates. One is 12 months and the other is 6 months. On the other hand the main difference between these two is not the sallary. The main differences are your rank and conditions.

In the shorter version you are a sergeant and you are inside the military 24 hours a day. You will probably sleep with 6 to 30 others in the same room. Some of these people will not be very clean. They will wake you up at a certain time in the morning and you will tidy your bed in a certain way. You will use the same bathroom with many many others.

In the longer version you are a lieutenant and it is like a job. You can rent a flat and live there and go to your job at certain times. If you have a car you can drive to the job. You can usually eat in nice restourants inside the military for very cheap prices. The longer version is much more comfortable.

natiypuspi is right. The 'y' in 'gelmeye' is a buffer letter* and the last 'e' in 'gelmeye' is the dative case**.

Why is there a dative case suffix in these words? Because these words must be an object of a verb that takes the dative case.

For instantance these are verbs that take the object in dative case:

-e çalışmak : to try x
-e başlamak : to begin to x
-e alışmak : to get used to x
-e gitmek : to go to x
The x is the object that takes the dative case.

examples:
gitmeye çalışmak : try to go
sevmeye başlamak : start to like
okula alışmak : to get used to school
sabretmeye alışmak : to get used to be patient
eve gitmek : to go to home
İstanbul'a gitmek : to go to İstanbul
yüzmeye gitmek : to go swimming, to go to swim

In the last examples above the first word is the object (e.g. "gitmeye") and the second word is the verb (e.g. "çalışmak").

As you see the object can be a noun (like okul) or a verbal noun (like sabretme). It doesn't matter.

These two sentences have the same transitive verb in the same noun state (dative). Dative case means "to", "towards".

"Doktora gittim."
"Yüzmeye gidiyorum."

In a sentence you will see the verb conjugated. For instance you will usually not see "sevmeye başlamak" but you will see it conjugated (with tense and personal suffix):

"Bugün Türkçe öğrenmeye başladım."

The 'e' in öğrenmeye connects the object "öğrenme" to the verb "başlamak".

If you have taken the words 'sabretmeye','sevmeye','öğrenmeye' from sentences then look to the predicate of that sentence. The predicate of that sentence (usually at the end) must be a transitive verb that takes the dative case.

It is pre defined what transitive verb takes objects in what noun case.

Examples:
"-i sevmek",
"-den hoşlanmak",
"-e gitmek",
"-den gelmek",

In other words, it is always "-i" case with "sevmek" and always "-den" case with hoşlanmak. Of course some transitive verbs takes can take objects in multiple cases. In this case the meaning changes.

Examples:
1. "-e gitmek" : to go to x
"Ankara'ya gidiyorum." "I go to Ankara."
"Yüzmeye gidiyorum." "I g swimming"

2. "-den gitmek" : to go from x
"Ankara'dan gidiyorum." "I'm going from Ankara.", "I'm leaving Ankara."

* We put a buffer inbetween whenever two vowels come next to each other. Y is the most common buffer. There are also s,ş,n buffers.

** The dative case suffix is -e or -a depending on vowel harmony. If the last vowel is a back vowel (a,ı,o,u) you continue with a back vowel, -a in this case, and if the last vowel is a front vowel you continue with a front vowel, -e in this case for dative case of nouns.

Consonant mutation has some irregularities.

On this page there is some information about this issue:
http://www.turkishlanguage.co.uk/conmut01.htm

Here are rules about consonant mutation:
1. Don't apply consonant mutation after a single syllable word.
2. Don't apply consonant mutation after nouns that were derived with the constructive suffix "t" (yakıt, barut, yanıt, yapıt etc)
3. Irregularities*:

çok > çoğu
teşekkür etmek > teşekkür ederim

* By irregularities I mean words that don't follow consonant mutation or that don't follow the two rules mentioned above.

A complete list on irregularities, which are not many, is unfortunately not available. There are limited number of irregularities but I don't know any source on this topic.

Greetings,
All the verb conjugations mentioned here so far are covered in winmekmak.

You can see the tense, suffixes, translations of the conjugations. There is no other program that can recognize a verb conjugation (*). Also winmekmak is for free. I have also worked on its last update.

1. open this page:
http://www.ipb.nu/winmekmak/

2. scroll down and find the link for the last version (0.2. Click on the link and save the program somewhere on your computer (for instance to "my documents" or "desktop".

3. After you have downloaded the program start it and press the F3 key on your keyboard.

4. Type a conjugation such as otursana, oturayım, otursun, etc. and press enter.

(*) Other programs work only the regular way. You give the infinitive and they tell you the conjugations.

For instance you enter "oturmak" (to sit) and they tell you "oturmayacağım" (I will not sit) is the future tense, negative conjugation for first person singular.

But with winmekmak you can enter "oturmayacağım" or any possible conjugation (with F3 key) and ask what it is.

Check a list of most sources on this topic:
http://www.turkishclass.com/forumTitle_6_6212
Compare #2 and #21 for instance.

"Şimdi yatma zamanı. İyi geceler."

If I had to pick a word for 'müthiş' I would pick 'extraordinary'. I believe this is the main translation.