What do the commas separating atomic events in a Relational Calculus formula means?
The notation is defined on page 17 and 18. The authors define a couple notations and state that they will "freely switch back and forth" between them.  This example doesn't seem super-consistent, I'd expect $Q(x,x'):\!\!-\,R(x),S(x,y),S(x',y),R(x')$ rather than an equals sign, but presumably it means $Q(x,x') = \exists y.R(x)\land S(x,y) \land S(x',y)\land R(x')$.

It doesn't really make sense to talk about natural joins in this context, as those rely on named attributes/columns. But yes, this query includes (equi-)joins. To be clear, the comma stands for conjunction.  Any joining is implicit in duplicate uses of a variable. In particular, if all variables were used only once in the right hand side, then there would be no equi-joins, even implicitly.  In SQL, the query would look something like:

CREATE VIEW Q
AS 
SELECT DISTINCT S1.Field1, S2.Field1 AS Field2
FROM S AS S1
INNER JOIN S AS S2 ON S1.Field2 = S2.Field2
WHERE S1.Field1 IN (SELECT Field1 FROM R)
  AND S2.Field1 IN (SELECT Field1 FROM R)


or to be more systematic,

CREATE VIEW Q
AS 
SELECT DISTINCT S1.Field1, S2.Field1 AS Field2
FROM S AS S1
INNER JOIN S AS S2 ON S1.Field2 = S2.Field2
INNER JOIN R AS R1 ON R1.Field1 = S1.Field1
INNER JOIN R AS R2 ON R2.Field1 = S2.Field1


or going a different direction,

CREATE VIEW Q
AS 
SELECT DISTINCT S1.Field1, S2.Field1 AS Field2
FROM S AS S1, S AS S2, R AS R1, R AS R2
WHERE R1.Field1 = S1.Field1
  AND S1.Field2 = S2.Field2
  AND S2.Field1 = R2.Field1