Sorry to hear about your cat. It sounds to me that the clear mucus is exactly that...mucus. Cats will shed mucus in their feces when the intestine is very irritated. There are mucus producing cells in the intestine and when an animal is experiencing inflammation in the gut they will produce extra material and it is often shed in the stool either with feces or on its own.
Vomiting and diarrhea in cats can be caused by numerous issues including:
1. parasites(like worms)- but often you see signs of the worms in the vomit or feces.
2. Stress and diet- changes in environment and diet all cause stress. Stress can harm the normal healthy bacteria in the gut and this lets bad bacteria grow and cause problems. Helpful things include feeding a bland diet like chicken and rice and to consider a pro-biotic. Probiotics help replenish the normal bacteria found in the intestines and help to make the gut healthy and stools solid. Some pet supply stores and vet clinics carry these. You may also able to find them on-line. They are non-prescription. My personal favorite is Purina's Fortiflora.
3. Bacteria and Viruses- Many different viruses can cause gastroenteritis (inflammation in the stomach and intestines). For severe diarrhea and vomiting cats may need to be hospitalized on IV. For milder cases cats can be given subcutaneous (under the skin) fluids and can usually be treated at home using bland diets, fluids (electrolyte solutions or water), stomach and intestinal coaters (like sucralfate/sulcrate) and probiotics. It can also be caused from eating high fat foods or other foods that do not agree with their system.
4. Foreign material ingestion (like tinsel, strings, elastic, plastic, etc). Some cats love to chew on goofy things. If these things are too big to pass through the stomach or intestines and get stuck they can cause vomiting. If these things are linear (like string or tinsel) they often get stuck in the stomach or the back of the throat and then travel down the intestines and cause the intestines to bunch up which leads to vomiting. Often, though these cats do not defecate.
5. Toxin exposure- Certain drugs, chemicals and other compounds can cause intoxication and gastric irritation and can lead to vomiting.
6. Pancreatitis- This is an inflammation of the pancreas. The pancreas is a glandular organ that digests fat. This can be caused by a high fat meal or for no reason at all and it leads to chronic vomiting and occasionally diarrhea. It can be severe and may warrant hospitalization. A simple blood test can tell whether or not this is occurring.
7. Other underlying health issues-like kidney disease, liver issues, etc. Many of these issues can be detected with simple blood tests.
In all of the above cases a trip to the vet is warranted. Vomiting can be serious as it can easily lead to serious dehydration. Also, if a cat is off food for a few days (even as few as 3) they can start to destroy their liver (a very strange process that seems to be localized to cats). They will take her temperature (to eliminate fever and likely viral infection) and may want to do an x-ray of her belly to look for foreign material that she may have eaten or do some blood work to be sure there have been no signs of intoxication, pancreatitis or underlying diseases. They can treat her by giving her fluids (either by IV if she is severely dehydrated or under the skin if the dehydration is mild) and they can give her anti-vomiting medications and gastroprotectants as well (so long as the more serious things like foreign material ingestion, viral infection and toxin exposure have been eliminated as possibilities). Also, I'd bring a poop sample in with you when you go to the vet (the nastier the sample the better the chance they'll find out what's going on....bring some of the mucus along too for them to look at). This will help them determine if there are any signs of parasites or bacterial overgrowth.
I hope this information helps and good luck with your cat!
P.S Here's a useful website on managing vomiting and diarrhea at home.