Calculate interbeat intervals
I splitted the question into smaller bits to allow more precise answers.


Question 1: you seem to not understand what spline interpolation is. The quoted sentence means that you need to use an interpolation function that guarantees the continuity of 1st and 2nd order derivatives. This is the case of cubic spline and a justification for their use.
Question 2: yes, I guess that there is a typo here and $n$ is $m$.
Question 3: it is an interval of sampling rates. It is not very clear though...
Questions 4/5: the paper is clearly not clear or informative at this point. What the authors do is to detect a heart cycle by looking at the 1st order differential points (not defined, but that are opposed to the number of local extrema). My guess is that these points are the zero-crossing of the signal derivative (computed from the spline interpolation, cf. question 1).


As an additional note, the paper is not really well written or informative. For example, Laplacian eigenmaps are poorly exposed, and prior knowledge is required to understand it. Maybe you should pick another reference paper in order to get started in this field.