The series starts out with a girl who doesn't think life is worth living any more. When she tries to commit suicide she is pulled into an alternate universe. This only able to happen because she is a throw away on earth. If she had purpose and a will to live she would not have been able to go to the alternate world. in this alternate world she meets a guy who will become her love interest. I believe he is a prince or an heir. There are several books in this series. I am thinking it was written in the 80's. At one point in one of the books the girl and a dog are passing through one of many parallel universes. This one happens to be ruled by dogs. Dogs are the masters and humans are kept on leashes. That is about all I remember. Any ideas as to the author or name of the series?
Your question has a lot in common with the Mode series of novels by Piers Anthony. It's a series that started with three books in quick sucession from '91-'93, but Anthony was unable to find a publisher for the final book until 2001. Points that are close:
The main character, Colene, is self-mutilating (cutting on her arm), and refers to herself as suicidal, but has not attempted suicide yet. Her situation is not great in a number of ways.
Her love interest, Darius, is from a parallel universe (of which there are many), and travelled more or less at random to find a bride that he can keep - his magic is to take in laughter and amplify it, sending it out to many, but this is draining on the source, which by custom is his bride, and he's worn through a few already. His offical title is Cyng of Hlahtar - basically a mangled version of 'King of Laughter'.
There are a near infinite number of parallel universes, each called a Mode. Darius' people have power to send him to Earth, but can't guarantee they can get him back to the same one. Since he wants Colene, they have to set up a Virtual Mode, which is like a cross section running through all the different Modes in between, letting you literally walk into a different parallel world every few feet. It works by anchoring 5 points, which represent 5 people with a need or want to be out of their current world, and Colene manages to grab hold of one.
There is indeed a talking telepathic animal named Sequiro that ends up being one of the other anchors, whom Colene ends up bonding with.
The plot involves all of the characters journeying through the Virtual Mode across the worlds, meeting and dealing with the other anchors, and trying to resolve the situations that got them to be an anchor in the first place.
Points that don't match up so well...
Sequiro is a horse, not a dog, although Sequiro's world does treat humans as servants and lower creatures.
Not quite writtten in the 80s, although it's pretty close.