Are there shots that can permanently alleviate my dog and cat allergies?

Yes. Immunotherapy - a series of injections of cat and/or dog dander under the skin over several years - will help you develop some tolerance to pet exposure. Some allergic patients respond better than others and a few patients not at all. Allergy shots alone are often not the complete answer. You may still need to get the pets off the bed, out of the bedroom, & perhaps out of the house.
Not really, but... Allergy injections, or immunotherapy, is the best way to control allergies - they make you less allergic, rather than simply covering up the symptoms like the allergy medications do. Many people find that their allergies go away completely with immunotherapy, but most people will see some of their symptoms return 3-5 years after stopping the shots. Long-term maintenance is an option, though.
Generally NO shots. For pet allergies-avoidance is the best treatment however that does mean you have not get rid of them. If pet allergy is mild then meticulous cleaning around the house is helpful, electronic air filters and keeping pets out of the bedroom!
No. No allergy shot is permanent. But you can get significant long-term relief with immunotherapy, specially to cat allergen.