HELP! Does this require a vet visit? *picture*

Here's a long story for you: I'm renting a room out from a woman who owns two cats, and I have grown quite close to them in the short time we've been together. However, this woman seems slightly anti-vet (I haven't gone into it with her out of respect, but it is my understanding that these cats haven't been to the vet since they were adopted). One of the cats seems perfectly healthy, but the other has a myriad of things wrong with her. For one, relating to this post, she has some strange black stuff in one of her ears. She is an indoor cat and doesn't scratch at the ear too much (not that I can see), however she does have a rather serious compulsive licking/biting thing going on with her, I'm not sure if this is due to some sort of cat eczema or what. Additionally, I've discovered some (very small, but still there) scabs around her ears and head. Sometimes it seems like she has a fever, her eyes get really dazed and she opens her mouth a little bit as if she's panting, but I don't have the equipment necessary to confirm. ADDITIONALLY, she was throwing up every day last week, but has since then stopped throwing up. My landlord was about to take her to the vet until she stopped vomiting, to my dismay. My hope is that if I can get someone to tell me if this ear situation is important enough to take her to the vet, then maybe she finally will. If it's just a normal thing or doesn't require a vet visit, that's fine, but I want to know so I can put my mind at ease. The first picture is of her right ear, the "healthy" ear, and the other two pictures are of her left ear, the "unhealthy" ear (sorry they aren't great pictures, it's impossible to get this cat to sit still!)
 



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It's possible she has allergies and the scratching is from that.  The ears are definitely dirty and should be cleaned.  She might have ear mites, if she's scratching her ears a lot and/or shaking her head.  I would encourage the owner to take her to the vet.

another thing I should mention: the cat wheezes and coughs a lot, but doesn't produce furballs

Mol Wattz said:

 
another thing I should mention: the cat wheezes and coughs a lot, but doesn't produce furballs

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That could be asthma which a vet would be able to diagnose and prescribe treatment for.

I know this cat probably has a lot wrong with it (as I mentioned) but I guess my main question is: is this black gunk enough to bring the cat into the vet...meaning if I push the issue with the cats owner, will she be mad at me for wasting money on a vet bill for something that didn't need to be treated?

The black gunk in her ear could be an ear infection. Ear mites almost always effect both ears so it's most likely not that. But an ear infection is definitely concerning. Maybe use that as a way to convince her to take the cat to a vet and then hope she brings up the cat's other issues as well. Or perhaps you could offer to take the cat in for her so that way you'd be able to talk to the vet yourself. This is definitely a tricky situation because it's not your cat but I'm glad the cat has you there to be her advocate. (Or "advo-cat"! Ha ha ha.)

There are some people who think taking their pets in to see a vet is a total waste of time and money.  It's kind of hard to change that way of thinking.  These are the same people who do not spay or neuter their pets.  Maybe you could get like a sample of whatever is in her ear and have it tested at the vet's?  Just a thought.