Our almost 4 month old son has recently started putting his fingers so far into his mouth that he ends up gagging. no signs of teeth ... normal?

That is very common: You probably will see a whole hand/fist in his mouth some day, and when he gets a hold of his foot, it may end up in his mouth as well. It is very common and so don't go crazy. It is harmless--virtually all babies do that to some extent. Enjoy the baby. He is priceless. Good luck.

Sure!: He may in fact be starting to teethe, but even if not yet, he may be doing this to satisfy his need for sucking on something, or simply because he is exploring his fingers. Lots of babies use their mouths to learn about their surroundings, hence everything - including hands and feet, too - wind up in their mouths.

Yes: Aversge age for teeth to appear is six months.