Sometimes cats vomit when they have furballs, and you have indicated the vomit has some fur in it, as well as some grass. Usually when the cat eats grass, it indicates the stomach is upset, or there is some sort of blockage, which could possibly be fur.
If fur has collected in the stomach from grooming, it can block passage of food, so the kitty may vomit back any food she has eaten. I am wondering if the food she is eating either is not setting with her, or if it simply does not have enough roughage in it. You might try changing foods to a normal cat food. Mix dry food along with canned and a lot of water. Many cats will vomit if fed dry-only, because it is simply hard to swallow - many will not even chew the dry food, but swallow it whole.
We ALWAYS mix dry/canned/water together, each and every feeding. The one time I played with the cats and fed them some dry kibble for little treats, little Opie vomited it back up, in a mess all over the hallway.
Truthfully, a cat of ten years isn't exactly a "senior cat", but closer to just past middle age. Our indoor cats have lived to about age 16, while the indoor/outdoor cats live between 18 and 20 years.
Good for you in taking kitty to the vet, which is the right thing to do. But, be sure she is pooping. Check the litter for regular poop. If you are seeing diarrhea and especially if it is watery, then she may have a blockage, and only the watery part can get past. If you are seeing no poop, she also may have a blockage. But, if she is also going outside, you may not really know what is going on.
Try giving kitty some milk. Cow's milk is known for going through a cat's system, thus possibly also passing a furball and making it easier for her to eat. There are also "furball remedies" that come in tubes. This is a gooey product made to help clear the system using oils. If she has fur lodged in her system, this may help.
Our cats love a plant called "Dracaena", (the green variety, not pink) which seems to work on clearing things from the digestive tract. We keep one of these plants, just for the kitties, and they trim it, often. Also, check to be sure the kitty is keeping her food down. For example, if she is going outside and vomiting her food, she won't gain any weight, despite eating every day. So, check to be sure she is maintaining her weight. If you need to keep her inside for a couple days, then do so.
Sometimes, worms can cause vomiting, diarrhea, or some coughing.
-Try a different food
-Mix dry/wet/water in a soupy mixture, each and every time you feed her
-Never leave food out for "self-feeding"
-Give some milk
-Use kitty hairball/furball remedy
-Buy Dracaena plant
-Keep inside and monitor her for a couple days
-Check for either diarrhea, or no stool
If no results, and kitty continues vomiting, try a different vet and be sure you let them know what you can and cannot afford. It is up to you to determine what tests you feel are O.K. and not O.K., because you write the checks.