Oh... I have several questions about this: 1. Would a native speaker still understand me if tried to pronounce all the letters? 2. Are there many such pronunciation exceptions in Turkish? 3. Are there any rules (of thumb) to remember such exceptions or do I have to memorize every word?
Of course. There are many natives, like myself, who do pronounce all the letters. But that's because of our dialect. Eastern dialects, like my own, tend not to reduce vowels. I sometimes say "unutmayacaım" sometimes "unutmayacaam", but only when speaking with friends and in family. If I need to be formal, I go back to "unutmicam".
Some. The future tense suffix is "-ecek" in written, but "-icek" in spoken, as you've noticed. "Değil" is pronounced "diil". The "R" in the present tense suffix -iyor is mute except for 1st persons: "unutmuyorsun" is pronounced "unutmuyosun". The "çt" group becomes "şt" in words like "Kaçta", (usually pronounced kaşta); (compare this with the Russian что-што). Words that have "ağı" is usually pronounced as a long "a". For example: "dağı" and "dağa" can both be pronounced as "daa" in careless speech. "Şuradaki daa (dağı) gördün mü?" For the same reason, "kâğıt" is usually pronounced "kâat"; and the accusative form of all the words in -ak; like "durak" --> "durağı" (easily pronounced as "duraa"). The Ğ is sometimes pronounced as "y" in words where it's preceded by an "e": "eğlenmek" /eylenmek/. This is not always the case, though. "Beğenmek" is just "beenmek" (at least in my speech. Perhaps there are people who say beYenmek.).
When in doubt, pronounce whatever is written. You can never be wrong. You can adapt your speech step by step.
I think your comparison with Russian is very appropriate. It has a rather complicated phonetics and one would sound as a foreigner while trying to pronounce words as they are written. Still, everyone would understand it. And, for a beginner, it is certainly not necessary to learn all those peculiarities. It will come naturally with the feeling for language.
Asla is more like "never ever". You wouldn't even dream of doing it; it's against your principles, or it's just simply impossible to do.
Hiç erken yatmıyorum. = I never go to bed early.
Asla erken yatmıyorum. = Never would I ever go to bed early. I consciously make an effort to prevent that from happening.
Out of the two, asla is less commonly used.
So you use asla like in Persian! It's just the hiç cognate is used slightly differently. In Persian we have to use "hiç" as a modifier of something. In this case, the thing being modified pertains to time, so we would have to modify what would be the Persian equivalent of the Turkish word "vakit ". The way vakit would be used is to talk about frequency (i.e. moments in time where you go to bed early). In short, the Persian way to say never would literally translate to hiç vakit in Turkish. This is why I was confused.
من هیچ وقت زود نمی خوابم.
"Man (Ben) hich (hiç) vaght (vakit) zood (erken) nemikhabam (yatmıyorum)"
edit: trimmed my comment
"Isn't that translation possible?" Yes, in the sense that it means about the same thing and is grammatical English. When translating, I like to mirror the original when possible. "I will never forget you" sounds so good and solid in English that I would choose it for my translation.