.....I've considered learning to play the piano, but each time that the consideration comes to mind so do some rather unpleasant doubts. The main cause of the doubts comes from my age. I'm 19 years old.....
I believe that age doesn't effect ones ability to learn the piano (unless we are talking about extremities in age, very young or very old) If you have a natural talent for piano then if you take it up when you where a child or an adult I really don't think there is much difference. I have taught young children who find coordination extremely difficult and I have taught old adults who learn new motor skills very fast. I think adults tend to be more disciplined to their approach compared to youngsters thus on average beginner adults I teach progress much faster than beginner children if they work hard on their practice method and consistently practice. Children tend to be fun motivated, adults tend to be achievement motivated. I think there is something to learn about seeing music through the eyes of a child and simply enjoy and have fun at what you do.
I really, really would like to be able to play the works of Liszt and Rachmaninov one day, but for some reason, I can't help but think that these aspirations could very well be nothing more than excessively wishful thinking.
They are long term goals, which are great to have and aspire towards. But you should develop some goals which are more immediate. Short and mid term goals are a prerequisite if you every want to make pathways towards your dreams. I advise you read books on goal setting and self motivation, I was drilled these books when I was young by my parents and never regretted it. Working without short/mid/long term goals all at once will retard your progress and make you founder about, which is very easy to do when learning a musical instrument. This is why a good teacher is important because they can set the goals for you, they know what you do not know and you simply cannot help yourself here effectively, unless you are truly auto didactic to your approach, in that case you wouldn't be getting overly upset over your long term goals because you should have realize the work load and pathway to achieve your aim. Even auto didactic's require to stand on the shoulders of giants and need to learn from those greater than themselves if they are ever to progress at an effective rate.
Question 1: Is my goal of being able to reach the skill level where I could play Liszt's advanced works (difficulty: Campanella and higher), despite the extremely late start feasible?
It is certainly possible however you need to break down your work to move towards that long term goal. This is the problem you face, you do not simply start working on these pieces now in hope that you will learn them, this is a foolish and slow way to go about learning the piano. You need to build your skill level so you can somewhat sight read these works and the technique and fingering required you can draw from past knowledge from other pieces. You also need to acquire a musical ear to be able to naturally understand the musical language and not forcefully recite the expression or rely on copying your favorite recordings (although there is nothing really wrong with this when you are learning but in the end we need to be able to find the expression ourselves and not always have to be shown it, something very valid for the other pieces you will be studying and not the ones you unavoidably have listened to countless times and already have been spoon fed the musicality).
Question 2: Have any of you ever known of any late-starters (19 and up) who managed to reach a very high level of playing? (It would be nice to hear some inspirational stories.)
High standard I would consider concert performing standard. Very High standard could include those who can play insanely complicated works which makes Liszt and Rach look like chopsticks, this constitutes a very small % of pianists and even at these standards society don't really care, it doesn't sell more tickets to a concert that's for sure!
I have taught a handful myself so I can vouch that they do exist (High standard late starters). Each of these students however never remained at beginner/intermediate levels for very long. They jumped grades fast, they had a natural understanding for music and fingering/coordination at the piano. Many of them came to me self learning at around an AMEB grade 5 level, in which I can consolidate their ability at that level then push them further often skipping to 8th with most of them.
Each of these students work disciplined and pretty much every single day. These are the serious pianists who pay good money for lessons and work hard using their teachers advice. They motivate themselves to practice even though they may make mistakes, they try not to get discouraged when in lessons they are corrected in passages they where set for homework which they thought they had conquered. They are very interested in their own progress and do not think about others, they do not let failures and problems get them down but constantly rise to the occasion and are happy to fail as many times as they need to until they finally win. The organize their work effectively so they constantly can see progress and know the occasions when they are slowing down.
Of all the late starters who end up playing at a high level I have taught each of them already have a natural musical ability. They have a high pattern observation IQ with music, this allows them to acquire good sight reading skills, they also have an intelligence to categorize technical skills they learn and notice it when it is being called upon in new pieces they learn. They all of course love piano music and have great passion for certain styles and select pieces/composers. These are the main traits I found from the late high achievers.
Question 3: If I were to start, do you have any recommendations as to what I should do? (Besides the cliche "find a good teacher", "practice with control", "practice often".)
If you are serious about your piano goals you will get yourself a teacher to set out the path for you to go to achieve your dreams. Yes this is the answer everyone gives but it is really important. There are just too many things that you Do not know that you Do not know when it comes to piano that without a teacher to start you off you will simply wander aimlessly about the place. Some people do this for a bit then get their bearings, in that case why not simply start learning music that interests you that is of an easier level. You do not have to worry about what technique it is teaching you etc, just learn lots of easier pieces and get them memorized and mastered. The more easy pieces you learn the more they become real natural routine for you to play and sight read. Then you will find that pieces of this similar standard that you have not learned before you can almost immediately play with mastery. Then you start pushing up your standard of sight reading. You keep doing this process until your Liszt and Rach start to become routine to read. But it is a slow process, it may take you 20 years before you get to it, but this is not important.
You will of course try to read and learn material too difficult for yourself, I don't think any pianist is innocent of that crime! It is however good to leave them aside for a few YEARS, then come back to them after you have trained yourself up, you may surprise yourself how much easier it is for you to read them. I constantly entertain myself doing this getting out old books that I grew up with that I never really look at. They use to be so difficult for me to play when I was younger but now looking back at them after 10-20 years sometimes, they seem like childsplay. You need to disconnect from these pieces for a long time however to really notice your improvement.
I loved Godowsky when I was younger and foolishly attempted to play his complicated compositions, it is not only in the last 4 years or so that it this material has become somewhat routine to learn and I am amused at how I can solve problems immediately where when I was younger I can still remember the difficulties I had (and the wrong fingering and ideas I see scratched onto the page reminds me of where I was going wrong). This is extremely educational for me to be able to compare how I worked 20 years ago with how I worked with today, the ancient pencil scratching is a reminder of where I have come from! But you need to let go of your long term dream and take it away from your mind, it can slow you down incredibly if you do not, put it aside and humble yourself by learning things that are closer to your level. Only once you humble yourself will you be able to become great.

All the best with your musical journey.