Among all these statements

We focus on #4 and #5 first. If #4 were false, then there is at most 1 true statement and at least 4 false statements. This forces #5 to be the true statement. However, #3 would have to be true as well, since #4 is false. Thus, #4 must be true.

If #5 were false, then the rest of the statements are true. However, #2 would then be false, since #1 is true. Thus, #5 must be true.

Since #4 and #5 are both true, #3 is false. This leads to #1 being false and #2 being true. In total, the true statements are #2, #4, and #5, or $\boxed{3}$ statements.

Suppose that #1 is true. Then #2 is false and #3 is true. Here, we have at least two true statements so #4 is true. According to #3, #5 must be false, but it is self-contradictory because there would already be 2 false statements. Thus, #1 must be false.

Accordingly, we have #2 is true, and #3 is false. As we now have 2 false statements, #5 is true and #4 is also true. So we have 3 true statements (#2, #4, #5)