What gear will they shoot with?
Is this their full time occupation?
Not sure I understand why these two items would be important at all?
Who cares what gear they have if you like the pictures? The final product is all that matters to the customer. If you like the final product that the photographer produced in other weddings, you will probably like the final product for your wedding. A photographer can have the latest, most expensive gear, but if you don't like their final product, that photographer is not right for your wedding -- regardless of the gear.
The same applies to the second item. Who cares if it is their full time or part time employment -- if you like the type of photographs the photographer takes? This is why it is so important to carefully review past work. A full time photographer can take photographs you don't like just as easily as a part time photographer... and vice versa.
There are so many more important considerations when selecting a wedding photographer. A wedding photographer is a business of which taking pictures is but one aspect the customer is paying for. Which gear they use and whether they do this full time or part time are not, in my opinion, high on the list.
Past performance is. And past performance is independent of gear and employment status.
Do you like the way this particular photographer shoots weddings? There is no such thing as a good wedding photographer. Only wedding photographers that are good for a specific customer. Not all wedding photographers the same and it is important to choose one that is right for your wedding. A wedding photographer that worked well for one wedding may not work well for yours. It is a personal service. That's the advantage of hiring a professional.
I think as time goes on, we will be seeing fewer photographers that can exist solely on wedding photography. I think the part time photographer will be becoming more common as the industry continues to be over saturated. I, personally, would not discriminate against a photographer solely on the fact they have another job (whether photography is the primary or secondary job).
I will discriminate past on past performance.
there are a couple relevant answers to the gear question - Gear is just too all inclusive but asking if they have backup systems (aka 2 bodies) is important. No one wants their primary camera to die on the day of a wedding, but, even with a top of the line camera sh!t happens. So does the photographer have a backup camera is a very important Q.
Full time vs part time - this can also be important, of course it does not truly make a difference - but - it may be a psychological reassurance - a full time, established photog whon't just take the money and run for instance. But also, turn around time. If your full time gig is photography your turn around times may be a bit different than someone who shoots weddings on the side while also working a full time job (40 hours a week punching the clock somewhere else means that's 40 hours of the week that they are not working on your wedding!)
As Chuck mentioned, gear and full/part time occupation play a role in the whole process. I know this because I went from a T1i, 17-40L, and 430EXii to dual 5Diiis with a third in the bag, tons of L glass, two 600EX's and upgraded by way of the 60D, 7D, and 5Dii along the way. My first wedding was decent, but I'm happy to say that I have come a long way in both talent and gear. I know that the gear is not everything, but it does reflect at least a partial amount of the investment the photographer is willing to make in their clients. There is a less expensive option to everything I own, but knowing that the 5Diii does better than the 60D (in my opinion) and the 85 f/1.2L ii does better than the 85 f/1.8 in portrait stills (in my opinion), tells me that if I want the best for my client, I buy the best. Gear, will definitely have an impact in the outcome of the work. Someone can always just buy the same setup as me without having to work their way up, and then it goes back to whether or not someone likes the portfolios they have to offer. We all have to start somewhere, and I covered expectations with my first wedding client extensively prior to taking the gig, but I also know that asking what gear they are shooting with is a valid question.
To answer your question about full/part time relevance, I used to shoot part time. I spent 4 years of undergrad and my first year at the college of veterinary medicine shooting part time. Exams matter. I hated having to put edits on the back burner, or having to rush through them to get them done in a reasonable amount of time, but if I had to study, I had to study. Even during the summers when I was working full time or part time in the winters, I had to be at work. It took my turn around time further than it would have if I was a full time photographer. Now, my wife and I have ventured into photography full time and have the capability of turning edits around faster, all while doing a better job than we had before. We have even hired a third photographer/editor, another editor, and an album designer. If I am backed up editing one wedding, another person can take it, I give the final proofing, and we can deliver in weeks instead of months. Yes, the part-time portfolio can and should look great, but my best work was done with May weddings, right after school, and just before I started full time in June. If I booked someone for Fall, I let them know the timing I would be working with. Having been there, I also find full/part time photographer status to also be important.
Is the couple personable, friendly, experienced, and do they take the kinds of images that emote the feeling the couple is looking for? Those are the questions that need to be brought to the forthright.
If a couple cares about the kind of 85mm lens being used, well that's downright silly and tells me they're looking for the wrong qualities in the photographer. I wouldn't shoot their wedding if that were the case. That example is much different from your broader point of "does the photographer have professional gear" - which is a valid expectation.
Further, whether the photographer shoots part-time or full-time is also somewhat irrelevant, provided that the turn around dates are discussed and agreed upon and the other aspects of their work are found worthy.
Surely, these points can be discussed in a sit-down meeting with a client, but it seems strange to expect a couple to care about what aperture the photographer's lenses shoot at.