I am looking for some advice.

My mother's house was wired up for an alarm system at one time using
security screens, door switches and some window switches.

For some reason, the wires going to the control box were pulled up
into the attic.

That's the easiest way to run the wires, especially if the system was
installed after the house was built.

Can I check each wire using a continuity checker and opening/closing
each door and window?

That should work. Make sure the line is good all the way back to the
controller.

One weak area has has two "doggie" doors that are big enough for
someone to crawl through.
(One goes from the yard to a outside room and the other from that room
to a middle room)

I can't see any way other than getting rid of the "doggie" doors.

I think you left something out of that sentence. I'll assume you mean
"...way to make the house secure other than...".

I always recommend _against_ installing pet doors. People fit though
lots of them just fine. If a person won't fit through, the burglar can
always fit extension tools through. Unfortunately, I'm not always
successful in talking people out of the installation, so I put it in
anyway.

If the doggie doors are installed in people doors, The cheapest
solution is to get new people doors. If the doggie doors are in a wall,
you'll need to add some 2x4 framing and wall covering.

--
Steve Bell
New Life Home Improvement
Arlington, TX USA