Why was I misdiagnosed with arthralgia? Can I start a claim for my Cervical Kyphosis with an uncharacterized discharge?

SPC (Join to see) As one who has actually worked in this field (VA benefits), both as a paid professional and volunteer, I can tell you there is some sage advice offered in this thread. First I will explain a few things. When determining service connection, the VA ONLY considers if the claimed condition was caused or aggravated by your military service. Having a preexisting or congenital medical problem is not a bar to benefits if it can be shown that the condition was made worse by your service. The fact that you were misdiagnosed on active duty may be a problem when not fought through VA as soon as possible following separation. See comment from SFC Melvin Brandenburg, time is of the essence. VA gives infinitely more weight to accurate and current diagnoses than something that can be shown to have been an erroneous diagnosis in your service medical records. See advice from SPC Celestino Alicea. Another factor the VA will consider is were you on active pay status at the time the claimed condition was incurred or aggravated. From what you wrote, I'd say the answer is "yes," you were on pay status at the time of your ruck march. They also consider if it was due your own willful misconduct. I don't see any indication of that here. Length of service is NOT germane to your claim, nor is your uncharacterized discharge. That matters to DOD, but not to the Department of Veterans Affairs. Those departments have different and unrelated standards, so don't get them confused. Your next steps should be:
1. Follow the advice of PO3 Donald Murphy and do not file with VA directly.
2. Google, or use the VA website va.gov, the location of your nearest VA Regional office.
3. With that information Google the contact information for a nationally accredited Veterans Organization (DAV, VFW, AMVETS, The American Legion, MOPH, etc.) colocated at the VARO. It is probably best not to go to the local Chapter or Post -- go directly to a National Service Officer.
4. Contact the the National Service Office of your choice.
5. File your claim with that NSO.
6. Follow the National Service Officer's advice on how to best develop your claim, some of which has been correctly summarized in other messages here.
7. Quit reading this and get started!

see a veterans service organization ASAP, and get documentation from medical providers that show there was no previous evidence of your conditions.

SPC (Join to see) I’d listen to PO3 Donald Murphy get an outside doctor to do your evaluation. I’d also check with your own doctors from when you were younger to see if any of that is in your record. My spine is a mess due to service connected issues and I had to fight them every step of the way.