This construction is very common in german but not used in english french spanish. I love using it because like most german grammar its so logical. I learnt it like this: Ich trinke gern Wasser, ich trinke lieber Wein, ich trinke Bier am liebsten, I like drinking water, I prefer drinking wine but I like drinking beer the best. This is used to say anything that you like / prefer/ like the most without having to use two verbs.
"lieber" is the comparative form of "gern". "Liebst-" is the superlative. So if "gern" is "like to (do something)", then "lieber" is "would rather to (do something)" when you compare two things. Here the literal translation is "I would rather like blahblah", but "would rather like" is just "prefer" in English.
I don't think there is a difference in the meaning, just that ich mag lieber is more commonly used than bevorzugen. Man kann auch "ich mag lieber Katzen" sagen, in fact I think that construction might be more common than "ich mag Katzen lieber". Disclosure: I am not a native or advanced level German speaker, but I have lived in Germany for five months.
There is example of "haben....lieber": http://www.dartmouth.edu/~deutsch/Grammatik/Comparatives/Comparatives.html
I'm not an expert, but I don't think people would say "mögen...gern" since "gern" is already "like to (do something)". "lieber" is "gern"'s comparative form so it doesn't just mean "like", but "rather", so I think it's ok to put "mögen" there. People do say "mögen....lieber".
"möchten" is not a verb, "ich möchte" is the conditional of "ich mag".
"Ich würde lieber XX [verb]" means "I would rather like to [verb] XX" (Example: Ich würde lieber Tennis spielen. I would rather like to play tennis."). So with "würde lieber" you need another verb in its infinitive form. "Lieber mögen" works differently: Ich mag X lieber [als Y]. = I prefer X [over y].
That's not true. There's an example here: "Ich habe Rotwein lieber" http://www.dartmouth.edu/~deutsch/Grammatik/Comparatives/Comparatives.html
Am I the only one who automatically read this sentence as "I like English lover"?