How to interpret 2 variables separated by a comma in chained inequalities

I would take it as c, that both $x$ and $y$ are between $0$ and $1$ and think that it should be unambiguous. I might have a worry in my stomach that it was $a$ and be alert to the possibility as I read on or check back to make sure. I would say b is wrong and should be written the way you did.

It is a convention for $0\leq x \leq 1$ and $0 \leq y \leq 1$ and is mostly to avoid typing it out twice. The comma is used this way in the equivalent statement $x,y \in [0,1]$ as well so it's consistent with that notation.

The overall integral of $f_{X,Y}(x,y)$ has to be one. Let's then compare the different assumptions.

(a)

The integral $$\int_0^{\infty}\int_{-\infty}^1x+\frac32y^2\ dy\ dx$$

is not convergent.

(b)

$$\int_0^{1}\int_{x}^1x+\frac32y^2\ dy\ dx=\int_0^1x+\frac12-x^2-\frac12x^3\ dx=\frac{13}{24}\not=1.$$

(c)

$$\color{green}{\int_0^{1}\int_{0}^1x+\frac32y^2\ dy\ dx=\int_0^1x+\frac12\ dx=1}.$$

So, interpretation (c) seems to be correct.