I had a surgery a while ago. If I choose not to med board this contract, am I able to do that later on? Can I also still claim it through VA?

First, you don't get a say in the Medical Board process. You will be referred to a MEB by your Primary care physician or attending physician. Then that is referred to a PEB (physical examination board). At that point, you should be assigned a PEBLO (PEB Liaison Officer) to walk you through the process. Even if you are kept on Active Duty, you can still make a VA Claim when you get out. I had a DA Hearing Waiver, I still collect from the VA.
It is in fact easier to get rated if you have military record of your injury/condition.

I hate to be the one to pop your perspective but med boards aren't something you opt into when you decide it's convenient.

Med Boards are initiated by your Primary Care Provider when it looks like you aren't 100% deployable or if you aren't checking all of the boxes to be deemed fit for service. It's not a testament of your work ethic but it is a bare bones opinion on whether or not you are physically/mentally fit.

Once the process has begun it will continue until it is complete. Generally you will be placed on LIMDU while all of the evaluations are conducted. This is when you are assigned a case worker and placed in the PEB (physical examination board). You will also be be assigned a PEBLO (PEB Liaison Officer) and then the package is sent to the actual med board for review. Then their response is sent back, reviewed and you'll either stay in service or you will be processed for separation/retirement depending on their decision.

Bottom line is that you are either fit or you're not and your Primary Doctor will be the one that kicks it off. You have little choice in the matter.

I don't know about the med board part of your question but as long as this surgery is in your medical records when you do leave the service, (if it is the same as back in 1992), a form will be filled out during out processing and sent to the VA.
After I left in July 1992 that is what happened and the VA contacted me to come in and be evaluated. With everything logged in my medical records, I was awarded 60% disability. I am not saying this happens in every case though.
I would highly recommend making sure your medical records are kept up to date and all information is in them.