Well, I've only been in since October, but had the same sort of feelings before Christmas - I didn't seem to understand any of it, didn't understand the explanations and didn't seem to be getting any better, and had trouble getting motivated. All that lead to thinking that I just couldn't do it.
It's quite a lot better now, I understand some small bits reasonably well (I'm not sure when that happened, but realised it in hindsight) and I have lots of things to work on, which helps - It's hard to be motivated when you don't know what you can possibly do next!
Although it does feel like as soon as I get comfortable with something, I get my feet pulled from underneath me and have to venture back into the big wide world full of lots of other things that I don't understand.
Although, interestingly, one of my colleagues has recently submitted his thesis - he says that he doesn't understand other people's work either, and can't understand how it took him three years to do.

Well, it sounds like a learning opportunity!

I saw experienced lecturers suffer from exactly the same things during my undergraduate degree.

Q&A sessions worked ok, volunteers were often in short supply, so some method of picking people was used. If you ask easy questions first, you will probably soon start getting volunteers. One lecturer used to go through the list of people that had missed his previous lecture 'to make sure that they weren't getting left behind', or some other such 'caring' reason.
I never saw 'Discussion times' work at all!
One of the methods for dealing with talkers was to say 'Sorry, but I can't concentrate on my lecture when other people are talking, so can you either stop your conversation or do it somewhere else?'
These sort of tactics shouldn't be necessary, but I'm afraid they are.

I must admit, that we had everything on handouts, but I know that differed by department.

There is a factsheet here about importing pets; there are a number of steps to go through, it will take a minimum of six months after vaccination, and will also be quite expensive.
http://www.defra.gov.uk/animalh/quarantine/factsheet/eufactsheet4_050907.pdf

Whatever happens, it will be good practice for this sort of job interview. I doubt that anyone else will care that you may have taught somebody else there.
I'm sure that the interviewers will be well aware that doing a PhD does not automatically mean you give the best presentations!
These things are stressful enough, without putting yourself under extra pressure to 'perform'. Just try to relax, do your best, and be happy with that, whatever the outcome.

Have a look at the research interests and / or publications of your potential supervisor and see if anything interests you that you could write about (maybe they have MSc projects advertised?). I was told that supervisors would be well aware that applicants wouldn't know lots about the subject, and should instead write more about the things that I had done and found interesting at undergraduate level and would like to explore further.

I am also starting my PhD soon. Although I feel a lot happier being a 35-year old PhD student than a 34-year old undergraduate!

I am not pursuing a PhD for the sole purpose of getting a job in academia, so I am not worried about the problems that may present when I am older. I have enough experience and expertise to believe that I could give it up and get a 'proper' job if I ever needed to, so I am happy enough to do it just because it is what interests me, and the pay is just about enough!