With so many methods of communication these days I'm sometimes unsure which to use.
Email. Always. Business communication is via email these days. Anything else is secondary and reserved for unusual or urgent matters. And by "anything else" I really only mean a phone call. There's currently no other accepted way to communicate professionally, unless a contact has explicitly told you to use another medium.
Last-minute rescheduling of an appointment definitely qualifies as urgent and might mean you call instead of email, but it depends on how short-notice this is and how responsive the company has been to emails. If your past emails were read or replied to within a matter of hours and no later than one business day and there are several working days between now and the scheduled interview you can opt for an email first. But if you have a meeting tomorrow or after the weekend you need to call them as soon as possible. If you email and don't get a response within 1-2 business days you need to call.
Apologising for the inconvenience is expected regardless of medium. The less time there remains before the appointment, the more profusely you should apologise and you should also acknowledge that asking to reschedule with little notice is not ideal and not something you'd do unless it was necessary.
Also, in an interview when they asked when I'm available I said immediately because, at the time, I had no job. Of course that doesn't mean I'm available at a drop of a dime 24/7 but I get the impression that they expect me to be.
It should. At least, that's what most employers and hiring managers think. And in some respects it's true. If you're a recent graduate with no job or other commitments then you are expected to be highly available. If you aren't that available it's up to you to let them know. Volunteering, vacations, prior plans or other hiring processes are all valid reasons for that.
Also note that some of this is due to the inherent power disparity in the hiring process. Potential employers can do things that candidates can't.
Perhaps there was also some confusion on the meaning of availability as it can refer to either "when can you start?" or "when are you available for follow-up interviews?".
For example a few interviews told me to be ready for a phone call the same day to see if I'm invited back for a second round,
That's normal. Even people who are currently employed can usually fit a phone call into their schedule, provided it's a quick screen and not an hour-long interview.
and then the second round is the next day etc.
This isn't usually a problem, unless it's because they want to fill the job ASAP, which might cause problems if you want to compare offers from multiple companies. They reasonably assume that you're highly available and it's equally reasonable for you to reply that you can't make that work and give them your actual availability.
To summarise all the above: potential employers can't know things that you haven't told them. Just be open about your availability and communicate clearly, quickly and professionally and you shouldn't have any problems with a hiring process.