Wow! You really
have missed a lot... There's not really a short answer to your questions, but I'll try anyway...
They're not fighting about Logan. Rory did an internship with Logan's dad at one of his newspapers, and at the end Bitchum told her she didn't have what it takes to be a reporter. Since Rory is apparantly a big baby who has never been critisized before in her life, she broke down, stole a yacht and dropped out of Yale to become a fulltime brat. Lorelei didn't think this was the best way to handle the situation and expected her very spoiled daughter to grow up - Rory didn't agree. Hence no Yale, no speaking to mom.
Sorry about the irritated tone, it's not directed at you but at the incredibly annoying storyline which should have wrapped up by now but is instead stalling absolutely everything.
