Not everyone gets warts. 50% of the infected do not. You may already have it and should be tested by your GYN. Yes, you can be infectious and not have warts. After he is treated, he needs at least 2 months of no sex to make sure the warts do not return. Latex condoms help recovery.
Generally, warts take 3-8 weeks to appear on someone post encounter. Some people can get them years later. Some never get them. Anyone who has the virus in their body is doesn't mean that you can infect someone 100% of the time. Some people become non-infectious. No one knows why. Clearly, if you have active warts, you are more infectious. Most people wait 2-6 months after their last wart is removed and no more visible warts appear before having sex. Some people wait a year. The use of latex condoms prevents more warts from appearing during the recover period. The medical community isn't in agreement on why some people get more warts or not. Some people think it has something to do with the strength of one's immune system. Cigarette smokers tend to get more warts.
Basically, there are 2 things you should be aware of. His health is very important and not getting more warts is better for his health. Sex is secondary compared to his good health. You should get tested and confirm that you have it or not. Not everyone who sleeps with someone who has it, gets infected. If you do have it, you should assume that after he recovers and is wart-free for at least 2 months, you and him can have all of the un-safe sex you want for the rest of your lives since you cannot re-infect each other. There is no ping pong effect. Clearance does not mean the virus leaves your body. It just means that the site of the infection is gone and that infecting others who he hasn't slept with yet may not happen.
Now you see why this damn STD is so confusing. There are no one-size fits all answer to every person who has it and the medical community are not in agreement with anything other than "once infected, you have it forever".