This is a "stage" common to year old babies. A one year old is exploring his world. He's found his epiglotis or his throat and is checking it out. This is a common thing to do. Also some babies and people have a stronger gag reflex which makes it impossible to not vomit if the person puts his fingers too far in to their throat.
I recommend paying as little attention to it as possible. He started this by discovering that he gags and vomits when he does this. He did it again to see if it still happens. And again until he's learned whatever he's working on learning. When the parent strongly reacts to the vomiting then there is another element to check out. When I vomit, mother gets anxious. Will that happen next time. If you keep changing your reaction he is more likely to continue checking it out.
This is very IMPORTANT! The baby is NOT consciously aware of what he's doing. He isn't thinking, "let's see what happens next time." I also suspect that he might even forget what happened the last time and in a way rediscovers his epiglotis.
I would quietly clean the mess up, while calmly talking with him about what has happened or reassuring him if he's frightened.
I would also put a plastic sheet over any surface that this is likely to happen. When it's naptime get the plastic piece (not a full sized sheet) before taking him to his nap. You might leave him in his diaper and put a water proof draw pad over his bedding. Perhaps you have some left from when he was newborn or later when you protected his bedding while sleeping or changing him.
Now to the bleeding nose. The air is dry this time of year. He has dried "buggers" in a dry nose. This is uncomfortable! To stop it apply vaseline a couple of times a day inside his nose. do this especially in the morning when he first wakes up and in the evening before going to bed. Another thing that will help is to use a humidifier. My daughter and I both have electronic humidifiers that put out a cool mist.
I was stranded hat her home this week when she started the humidifier because all of our noses were dry and uncomfortable. She has a small 3 bedroom apartment and the one machine in the living room humidified all the rooms.
Here is a far out idea. Could the gagging and nose picking be related? How is he gagging himself. I assumed he was putting his fingers in his mouth. I have an adult friend who could gag herself just by thinking about it.
Perhaps his food isn't masticated (slushy) enough and hits his epiglotis combined with an easy gag reflex.
Sometimes I gag when whatever I've eaten doesn't feel good in my stomach. Sometimes I feel enough nausea that I think that if I could only stick my finger down my throat to make myself vomit I'll feel better. Many times I've experienced feeling much better after vomiting. Something like this could be going on tho it does seem unlikely.
If he's not putting his fingers in his mouth and seems to just spontaneously vomit it is likely something else is going on. I've read that babies will not eat too much. I've found that to not be true and when my cousin's baby ate too much she vomited. I had assumed it was OK to keep feeding her as long as she kept eating. Her mother said I'd fed her at least twice as much as she usually ate.
If the stomach is full, bending a baby in the middle or tipping them head down or rough housing with them can cause vomiting. The flap at the top of the stomach that keeps food in can be weak. I've experienced this with both of my grandchildren.