I can only really comment on my experience (and the experiences of the people who work with me in lab).
I work in a lab entirely composed of undergraduates. We are at a liberal arts school with no graduate schools in the sciences. We do tend to have one full time lab technician (usually someone who just graduated). My PI is also considering taking on a post-doc but hasn't decided.
Our lab has a specific focus, a big "question" that has over the years developed into many questions. Generally speaking we studying how a particular specialized organelle is formed in C. elegans intestines, and what the organelle does. That is an amazingly big question (though it might not seem like it). The question has gone in many different directions leading to many different projects.
In my time in lab, which will be 3 full school years and 3 full summers when I graduate in May (and I work ~40 hours a week regardless of summer or during school) I have worked on maybe 4 "projects". I wouldn't call them experiments because they were composed of many experiments. If I were to list experiments I've done (Where I've had a question and answered it) it would probably be over 100. In terms of bigger "questions", it is around 4. As an example I spent about a year studying how one particular gene was involved in the formation the organelle (genetic studies with other genes we have previously studied, and characterizing the phenotype) as well as addressing whether other genes that encode similar proteins play a role (the gene encoded an ABC transporter, so I screened other ABC transporters and actually found another one that had a similar phenotype). That involved MANY experiments, but it was all around one project, which was itself just a piece of a much bigger project.
Most "big questions" aren't going to be answered in the four years you are in undergrad...you never really "finish" anything (very rarely). The best science is the kind where when you get your result that gives you an idea of the next 10 things to do and so on and so forth.
I don't know if that answered your first question but that's about what I could say. If you had any other questions I could try to answer.
As for number 2 I don't really think it is important. If you put the time in, and your PI can attest to your commitment to the research I don't think it is necessarily important to have one big project for your entire time, versus jumping from little thing to little thing. Though I think it is more PERSONALLY satisfying to have big things that you "complete".
