ILLPN
The blood test to find cause of neuropathy is a set of blood tests recommended, it is generally called a "neuropathy panel", and your neurologist will know. Perhaps you already had the test. And then there are a long list of all kinds of blood tests for further workup. I find them all very complicated. Sometimes test come back negative, they just have to try over again, and there are hundreds of different complicated bloodtests, mostly to do with immune system. and autoimmune related. There are many autoimmune disorders that causes severe neuropathy. Do you have joint pains?
The ting you mentioned about burning of lips. That was one of my first symptoms. I got suspicious it had to do with an autoimmune thing.
( I belive this burning lips symptom is from the small fiber neuropathy.)

You asked : can one have all three neuropathies.
The answer is yes.
You can have peripheral neuropathy, sensory and motor and you can have long fiber, and small fiber neuropathy. You can also have autonomic neuropathy. All at the same time.

I hope you find answers, and good treatment.
The investigation of underlying reason for your neuropathy can, as Glenntaj said, often take a very long time. And it can be very frustrating.
Do not give up. If an underlying cause can be found, ant treated, chances are better. Ideopathic neuropathy (without established underlying cause)
is treated with symptom management. There are some good treatment mangament methods, nutritional, supplements and prescripition medications.
The meds often used are some antidepressant meds, they are good in targeting nervepain. They are often used in a combination with antiseizure medication such as Gabapentin or Lyrica.
Do you have a rheumatologist? Have you had diagnostic tests and X Rays for arthritis? Have you had MRI s of your spine?
Have you had EMG and NCS (nerve conduction study) of your extrimities to find possible entrapment that can cause peripheral neuropathy.
Herniated discs any where on the spine can cause pressing on nerves.
Arthritis, both Osteo and Rhematic, can also cause neuropathy.

Do you have any compression fractures? Do you have Osteoporosis?

I will recomend this book, "Peripheral neuropathy" by Dr. Norman Latov You can buy it on line
Best wishes
Synnove