From your description, all three courses look almost identical to what I took when I was undergrad. So yes analysis 2 will be more difficult and a lot more work like it demands more time you know reading, re-reading, and then re-re-reading theorems and proofs and lecture notes and the coming up with your own proofs and writing up the homework. And I think I agree with what you heard. Now that I think back, from all of the undergrad math courses I took, analysis 2 was the hardest. Number theory will be a piece of cake.
But remember being the "hardest" class doesn't mean that it is "hard" or impossible. In addition, the load also depends on how many other classes are you taking next semester. If these are the only two, then yeah for sure the load is doable. Otherwise if you want to take like five classes next semester including these two, then you should reconsider perhaps.
I never took both the same semester. I actually finished analysis sequence earlier and number theory was at the end right before graduation. But both were taken with a full time (3-4 courses every semester) load and I don't think it was that bad. Analysis took a LOT of time but number theory was like the easiest class I ever took. And FYI, this is only true for intro to undergrad number theory. It gets incredibly insane very fast if you ever want to take more number theory.
Royden, "Real Analysis" Part I + first few chapters of Part III.
Folland, "Real Analysis: Modern Techniques and Their Applications"
Cohn, Measure Theory
Since you mentioned baby Rudin, first half of big Rudin ("Real and Complex Analysis") is actually not bad, but is perhaps a bit unmotivated when you first approach the subject.
Best Answer
A first approximation is that real analysis is the rigorous version of calculus. You might think about the distinction as follows: engineers use calculus, but pure mathematicians use real analysis. The term "real analysis" also includes topics not of interest to engineers but of interest to pure mathematicians.
As is mentioned in the comments, this refers to a different meaning of the word "calculus," which simply means "a method of calculation."
This is imprecise. Linear algebra is essential to the study of multivariable calculus, but I wouldn't call it a calculus topic in and of itself. People who say this probably mean that it is a calculus-level topic.