Newbie Tag Heuer CY2110 Jo Siffert watch questions

Hi, I am a newbie to the forum so hopefully this is not inappropriate or the wrong place. I just purchased a used one for my son's upcoming graduation (lucky boy!). It appears to be in very nice shape. Some question I have so I can ensure it is working properly and well setup for him.

1) Should I take it for service? If so what what should I get done?
2) I am unsure what the pushers do. When I push the top one the 60 second chrono (on right) stops as well at the main watch second hand. That seems weird to me that it stops the main second hand. I can not get the 30 second chrono on the left to do anything but have not fiddled too much with it.

Sorry I can not find any information on the above so thanks in advance for any advice. I may look for a leather strap.

Many thanks. You are exactly correct on how it works! The 30 minute register is as you state. I had thought it was seconds. This was super helpful and I really appreciate it. I tried many searches and found nothing. He is super stoked as it took a bit of time to find this model. Thanks again!! Off to a watchmaker.