Confusion about Fubini's theorem and when it does or doesn't apply

Question

I've been reading into Fubini's theorem (rectangular regions) quite extensively over the last few days, and I've gathered the following:

1. For Fubini's theorem to apply, the function must be Lebesgue integrable - that is, the integral of the absolute value of the function over the rectangle must be finite, and the function must be measurable (though practically any function that can be described is measurable).
   2. All Riemann integrable functions are Lebesgue integrable except for conditionally convergent improper Riemann integrals.

So, how does Fubini's theorem work for these conditionally convergent improper Riemann integrals? Can you or can you not switch the order of integration?

Also, if you were integrating over a rectangle which was infinite in one or both dimensions, is that still a rectangle and does Fubini's theorem for rectangular regions still apply?

And lastly, how does Fubini's theorem for improper integrals tie into all this?

Thanks.

PS: I'm quite new to all this and I'm struggling to piece all the different strands together. I have no knowledge of measure theory, so I would appreciate an answer in simpler terms.