You need to realize that this is all new for the dog.  He doesn't know yet
what you expect, and the only way to find out is to try something. He needs
a lot of affection and lots of praise when he behaves as you want.  Terriers
can be a lot of fun.  Our little terrier mix was always looking for a way to
join in when the kids were doing something.  She had a serious tennis ball
addiction and taught herself to roll a ball down the stairs, go after it,
bring it up and roll it down again. She figured out to alert me when the
phone rang and I was doing something noisy like laundry or vacuuming.  When
she developed congestive heart failure and needed medication twice a day,
she would remind me if I forgot the time.

We had two big old cats when we got her as a pup, and I think they made her
an honorary cat.

Your dog may not be used to being in a cage all alone all night.  If he only
started barking a few times, he's doing a lot better than a lot of dogs.
Personally, I feel more secure with that little alarm looking out for the
household.  A few barks at a strange sound is good as he learns what sounds
are strange for your environment and gets used to the usual sounds. Of
course, you don't want him to keep on barking or to learn that the way to
get attention in the middle of the night is to start barking.  This may be
what you are teaching him by responding.  I suppose the easiest way to keep
him quiet at night is to let him sleep with you, but I would be surprised if
the cat would agree to that!

I think the classes would be good for both you and the dog.  A talk with a
pet behaviorist may help you understand how to meet the dog's needs while
meeting the needs of the cat.  You can change some behaviors, but remember
that the dog is a dog and you can enjoy him for his personality just as your
cat is a cat and you enjoy her for hers.
Annie