leak before shutoff valve and HOA want me to pay for it

1. Would you think that leak is the homeowner responsibility?

Possibly. Was the leak caused by something the tenant or a tenants guest did?
What do the governing documents specify?

If the documents specify that pipes on common area is the Associations responsibility and the break was on your property, then it may be your responsibility.

2. Assuming that it's homeowner responsibility, can HOA call for the repair without notifying homeowner and send them the bill?

In an emergency situation (and a water main leak would be an emergency) the Association typically has such authority. Read your governing documents to be sure.

3. I'm not agreed to pay but I guess I will have to. Is there a way to fight to get the money back after I pay?

Write on the check that you are paying under protest and consult an attorney (as you will likely need to go to court to have a ruling).

Another option is that you contact the Association and specify that you are disputing the charge and ask for an itemized bill.
Then call plumbers and get written estimates on what they would have charged to do the same repair.

Keep in mind that labor costs for an emergency is typically higher then for a job that they can schedule for their convenience.

The rule of thumb of water repair. Anything between the house and the main is the homeowner's responsibility. If it's between the main and the road, it's the Water department (HOA in certain conditions). Now each HOA is different. Not enough details to determine your HOA, Water department, or homeowner responsibility for sure. That is because not sure how/who pays the water bill.

Our HOA we used to pay for water in our dues. When a line broke the HOA was entirely responsible for repairs. Which got extremely expensive. So they changed it so that everyone has separate meters and owners are now responsible. The previous set up allowed us to turn off the water for not paying your dues. The new separate meters we no longer could. That is also when the house to meter was owner and meter to street Water department. The HOA could/would contact the owner or water department if a leak occurred. However, repairs they did not pay.

The HOA may have been in their rights to have made you pay the repair bill. You have to keep in mind this... If the HOA pays your bill it means ALL the owners are paying your bill. The HOA is ONLY funded by it's owners for it owners. It may be that the HOA members don't think this is a responsibility of theirs if a member incurs leaks. If that is the case, then it's agreed that individual owners pay.

The HOA could hire and fix this leak in an emergency situation. A water leak isn't something you can choose to fix on your timeline. We had a water leak so bad it was flooding several yards. The water department HIGHLY frowns upon not addressing leaks ASAP. The HOA may not have had a choice but to have fixed it ASAP. As you were not there to confirm how bad it was nor what damage it caused, the HOA took matters into their own hands. Which they did incur damages which they are in their right to minimize. The court will look at it as such.

I would pay the bill and move on. Next time set up a better communication to address such emergencies.

As others have said, read your governing documents to see who's responsible.

But also, What is your HOA's policy about billing Owners? This also should be in your documents somewhere, maybe your CC&Rs, your bylaws or in some sort of Rules & Regulations.

In CA, we Boards of Directors/HOAs must send a notice to the Owner calling them to a hearing before we can bill them for repairs or damage. That way, the Owner can ask questions, etc. Then the board votes on a decision in private. So, if your own docs don't have a written policy on this, maybe GA statutes do.

I have some question about the protest. Do I need a lawyer to bring this case to a small claim court?

Call your insurance company and file a claim. Sudden and accidental leaks are often covered. A good insurance adjuster can check your documents and help you sort this out, and arrange to pay for the repair.