Translation:The mother of Carmen has a new telephone.
In Romanian, when you have female names, the genitive is formed by adding in the plural form the letter "i". But in all the cases the names are ending in a vowel. Carmen is the only female name is not ending in vowel. In male names you put the "lui" before the name (to form the genitive). But I am not sure if this sentence is right. I am not a native speaker. I am waiting for someone native speaker to explain us that.
No, you cannot because of the rule mentioned above that the female name Carmen does not end in a vowel, so it is treated like a male name (e.g., "Mama lui Andrei.") There are other Romanian female names like that, but they are very rare (e.g., Catrinel).
You could, however, say "Mama Anei/Ioanei/Oanei/Andreei/Mariei/Ramonei/Simonei..." So, for Romanian female names that end in a the last letter is replaced with ei.
For Romanian female names that end in "ca" it's a bit different: "Mama Ancăi/Monicăi/Ralucăi..." Same for "ga." "Olgăi" came to mind.
There could be exceptions. I also read somewhere that for foreign female names you're supposed to use "lui" regardless of the ending.
Because even though Carmen is the name of a woman, it is treated like a man's name in this case, so we use "lui" before the name. I do not know the reason behind it, that's just the way it is. You'll notice in my previous explanation that for female names ending in "a," "ei" is appended to the name and the "a" is dropped.
If you said "Mama ei, Carmen, e simpatică." that would mean "Her mother, Carmen, is nice."
Yes, "ei" is the feminine equivalent of "lui" in certain cases like relative and absolute possessive pronouns. "ei" ("her/to her") is the genitive/dative of "ea" ("she").
It's Mama lui Carmen and not Mama Carmenului because you don't own people pretty much. Also, Carmenului just doesn't right. Names just don't always inflect. Some do, some don't. If you're seriously learning romanian, you'll find a pattern. Being a native speaker, i just know if it sounds right or not. Sorry, kinda not helpful but ¯_(ツ)/¯ ^^