If your camera allows you to switch modes by all means try them out. The bottom line is not what mode a picture was shot in, but what the final image looks like. How you get there is up the the photographer, but the image is what you will be judged by.
No, not all professionals use manual mode, but I would suspect that all real professionals know enough to use it effectively. Using aperture or shutter mode are decisions that need to be made depending on the situation being photographed. For instance if you are shooting people moving on a well lit stage and the camera is back 40 or 50 feet you won't be as concerned about depth of field, so you could use shutter priority and let the camera decide the aperture for the changing lights. At that distance even a large aperture (smaller f number) will carry a relatively large depth of field for short and normal lenses.If you find the camera can't make the exposure and is running out of aperture you would use a higher ISO.
See this for a depth of field calculator.
Manual mode should be the first thing you learn if you are serious about learning the craft. Since someone is doing you a HUGE by taking you as a protege, do her the courtesy of following her instructions, your experience will be more the better for it.
The shutter speed and aperture settings are creative tools, you use them to get the effect needed for the job. If you are using a priority setting you may fail to get the depth of field needed or the motion affect needed for the image you are supposed to be providing the client. There is no excuse for poor results if you present yourself as a professional and are taking money for your work. You need to know your equipment and the process enough to insure that the results are what the client expects. If you don't give them what they want, they have the right to ask for a re-shoot at your expense.
So the best thing you can do for yourself, if you plan on becomming a professional, is to learn how to make correct exposures (read the histogram) by making informed judgments on the aperture and shutter speed. ISO is something you shouldn't have to deal with much. You only need to change it in low light, and then you set it high enough for the bulk of the job. The last thing you need to do is learn to focus manually. Camera sensors will not always pick out the correct point of focus, but the eye will. Having a depth of field calculator will help here.
In the last 5 years I can think of maybe two shots I took using a priority setting. I have used auto focus on moving objects or in low light situations. But in low light I used it to set the focus where I wanted it, then shut it off while I framed and took the shots.
Ultimately, IMO, priority modes are crutches for people who can't be bothered to learn to use the tools correctly. Also, there is so much info on the web about the technical stuff, and sorting it out is difficult when trying to determine who to listen to. Many of those write about it don't have the best of education in the medium and tell you what works for them. That doesn't mean it will work for you. The best educational site for the technical aspects that I have seen is Cambridge in Color, they have a great set of tutorials.
There is also a great product called 123DI, a download tutorial that gets as detailed as you want to be or not. It is arranged in a way that you can get just the dummy level, more detail, or the techno-geek version of the information. He also goes into basic post processing using various popular programs. When I needed to make the transition from film to digital after 30 years (18 years in the industry), 123DI did the job.
It's hard to say, but my guess would be yes. Manual gives you so much more control over your camera. If you haven't learned yet, ask your mentor (how cool to have!) to show you. It really makes a difference! :)
Mostly, but not always. And mostly I use Raw-file - the camera does not the job for me, as sharpening/contrast etc, but I do it myself on the pc, - the file will not be compressed as all other files, not before I am happy with it, and save it into jpg.
If she has taken you under her wings, I suppose that is the way she can learn you most. As you say, you are an amateur, and luckely you get support......
To understand Manual, it is fine to try out and understand what choices the camera make when you set, for instance S (time) or A (aperture) too. (And +/- EV)
It is all about getting the right light to exposure.
A= bigger or smaller opening in the lens that give light, the bigger opening (small f-nr), the more light, but less dept in the pic.
S= shutterspeed can freeze a motion in high number, when it shoot fast - then you get less light.
These things you can train yourselves anywhere, anytime, to learn. The pics do not have to be a motive you want to save, just to learn from. Tose, nose, chairs, lamps etc......
When you have found out the bottoms and feel fine with the different choices, and start to understand how they work, you can start with A, then M - same pic, and S, then M - same pic, and compare....Whithin 2-3 weeks daily training, you will be a much happier photographer.
P.s.: And when you are going out shooting, as you have done until now, use your Auto, and have FUN!!!! Never forget - have fun!!!!
For me, there are certain types of pro that would always use M, being studio, fashion photographers as camera's don't meter for studio lights and so Manual mode is the only way to shoot. Most other photographers most likely use the mode (Av or Tv) that suits their chosen genre best. A sports photographer most likely shoots in Shutter priority Tv mode (just a wild guess, by no means am i stating this as fact). I don't think there is a right answer to this as a general question. Manual works for some and not for others. For me, it only works in the studio, or still life images where i've controlled the light myself. In the main i use Av and exposure compensation.
As for the question 'what is a pro', the answer for that to me is someone who makes their main income from photography, and lives off that income. I work full time, earn a wage, pay a mortage but not with photography. I have been paid for my photographs, I am doing a few weddings next year and charging decent money, but I am NOT pro, as i do not make a living from Photography. And again, in my opinion, pro means nothing more than that. It does not mean good, no more than amateur means bad. Anyone who has browsed images for a while here will know there are people here claiming to run successful photography businesses (and i'm not suggesting that's not the truth) who are nowhere near as talented and creative as some of the amateur's here. Making good money is well documented in almost any photography book to be 10% good photography and 90% good business sense. If good photography alone could secure you good money, there would be many rich photographers at MSS.