Fairly new myself...
I often tell people I started in 4th grade. That was the year I wrote 4 books in my "Archie The Ant" series, all 4 being illustrated by my friend Leo. I wrote them, bound them, and began reading them to the kindergarten, first grade, and second grade students. Eventually they were just added to the school library.
It was in 2003 when I finally sat down to seriously write a manuscript. I had been teaching 4th grade for a few years, I had just completed my graduate degree, and I had been saying for a long time "I want to write children's books someday". Finally, I was cleaning the basement with my father one day, stopped short, and wondered "What am I waiting for?". Thus, I began writing my first manuscript, inspired by my grandfather and written as a tribute to him. That was 3 years ago. After a series of rejections from publishers, I used a POD publisher so I could have it in print for my grandfather to enjoy while he had his health. Through the use of PODs, I have sold about 600 copies, but I am still waiting to be "published".
Okay... I am starting to rant again...
So, question #1, I would say I have been writing "seriously" for three years.
As to how I kept my hopes up through rejections, that is a mixed response. I really believed in the value of my novel and I was getting a lot of great comments and reviews from readers, so I let those good responses overpower the rejection letters. I also kept believing in my own work. In a way, however, I did let the rejections get to me. When the book's life with POD began to hit some real snags recently, I was not too motivated to try sending it to agents or more publishers. Why? I am guessing the rejections did something to thwart the enthusiasm and confidence. It has come back, though. I never actually sent my manuscript to an agent, so I plan to start providing some with my work, and maybe even go direct to a few more publishers, too.
I think what you will find is many of us are in the same boat... searching for ways to keep believing and keep hoping that the good news will come in. One reason I was so eager to join this group is because I think the community can help support each other. Even if someone in the group does not feel like they are in the same boat, it probably means their ship has landed and they have plenty of tips on a smooth journey.
Okay...Okay...I really need to stop ranting and get to sleep. Hopefully that at least answered your question, and maybe some of the extra stuff was helpful, too.
All the best to you and yours!