"Beni neden sevmiyorsun?"

yes and yes

No, that word order doesn't make sense in English. "Why don't you love me?" would work though.

In questions, the subject goes immediately after the auxiliary. "You do not love me" -> "Do you not love me?". "You don't love me" -> "Don't you love me?". Even though "don't" historically derives from a contraction of "do not", and most English textbooks will still describe it that way, it (and all of the -n't forms) behave like single words. In effect, -n't functions as a kind of negative inflection on auxiliary verbs.

That is not correct English. You must say "Why do you not love me" or "Why don't you love me" in English :)

Please be aware, however, that most English-speaking people will understand your question even if you get the order wrong. But it will sound foreign to them.

In addition to all the other great answers above, I just wanted to add one thing. This word order does appear in English but only when the question is embedded in another sentence.

For example:

Please tell me why you don't love me. I don't understand why you don't love me. I want to know why you don't love me. I have no idea why you don't love me. Did you tell your mother why you don't love me? Do you know how much it hurt to hear you explain why you don't love me on national television?

In all of those cases, the highlighted part of the sentence is not actually asking a question. In the last two example, the whole sentence is a question, but the "why you don't love me" part is not asking anything. It essentially stands for "the reason that you don't love me."

Note that the order of the verbs is reversed in English questions. The auxiliary part of the verb (do, have, will, etc.) precedes the noun in English questions:

Tom has studied every week. Where has Tom studied?

Sue will not like the cake. Why will Sue not like the cake?

Beth did go to the store. When did Beth go to the store?

yes it has no meaning

if you have a question word (neden, in this case), you cannot use mu-

If I recall correctly, mu/mü/mı/mi are ONLY for making questions that have "yes" or "no" for an answer.

"benim çirkin olduğum için mi"

The former is for specific direct objects. The latter is for indirect objects and motion "to me." They are most certainly not interchangeable. :)

beni = accusative

bana = dative

Which one you use depends on the verb and what you're saying. Generally, the dative case corresponds more or less to "to".

Onu bana verdin. = You gave it/him/her to me.

Beni ona verdin. = You gave me to it/him/her.

Some verbs require a somewhat unexpected case, although they usually don't seem too unusual.

Beni gördün. = You saw me.

Bana baktın. = You looked at me. (Kind of like "you looked to me")

Bana yardım ettin. = You helped me. (Kind of like "you did help to me")

Bana sarıldın = You hugged me. (Kind of like "you hugged to me")

Benden nefret ettin. = You hated me. (Kind of like "you did hate from me")

Any good dictionary will indicate which cases to use with verbs, often by showing one form of the suffix in italics or brackets, like this.

görmek (-i) : to see (sb/sth)

bakmak (-e) : to look at (sb/sth)

yardım etmek (-e) : to help (sb)

sarılmak (-e) : to hug (sb)

nefret etmek (-den) : to hate (sb/sth)

No. You can use just like these: "Beni artık neden sevmiyorsun?" (correct but not common), "Artık beni neden sevmiyorsun?", "Beni neden sevmiyorsun artık?" I can't explain the grammer state but you can't use "artık" between the root question "Neden sevmiyorsun?"

Will you go home? - Eve gidecek misin? Will you go home anymore? - "Artık eve gidecek misin?" or "Eve gidecek misin artık?" or "Eve artık gidecek misin?" (not common)

Don't you do this work? - Bu işi yapmaz mısın? Don't you do this work anymore? - "Artık bu işi yapmaz mısın?" or "Bu işi yapmaz mısın artık?" or "Bu işi artık yapmaz mısın? (not common)

Mostly we are using "artık" at the beginning or at the end of the sentence.