If you ring up and request an appointment, you must be offered one.
However, a lot of problems can be addressed over the phone and telephone consultations are fairly common. Bottom line is if you ring up for an appointment in no way should you be refused. Dont get into the symptoms, problems or whatever, simply ask for an appointment or a double appointment if you have more than one problem to discuss. If you don't get one for whatever reason, complain to the practice manager.
As for home visits, yes Doctors are reluctant to do these for anyone who is able to get to the surgery due to demand. And this seems fair enough imo.