If you are attending a club launch, then you need to check with the club about the rules for the launch. Some clubs may just use the FAA definition for Class 1 to limit the kinds of flights allowed — they are only concerned about whether they need to get an FAA waiver. While other clubs may also limit the kinds of motors allowed — they don’t want to be checking high-power certifications or making judgements about the edge-case motors, or they may have landowner or other restrictions.
Most clubs I’ve flown with have followed the stricter set of rules for launches that are not their HPR launches. They don’t want high power by any definition, including both FAA and NFPA. So no Class 2 rockets that would require an FAA waiver — rockets must be under 1,500 grams, and propellant must be under 125grams. And no high-power motors that would require a high-power certification — average thrust must be 80N or under, total impulse 160Ns or less, no sparkies.
I think it would be great for more clubs to have Class 1 launches and allow the high power motors under 125 grams propellant, because the club would not need a waiver, but could still allow some Level 1 HPR flights, including cert flights with rockets under 1,500 grams and H motors under 125 grams, like the H135, H115, etc. It would also allow high-thrust F and G motors, like F100 or G125. And it would allow sparkies, like G75 and G72. Unfortunately that’s kind of rare.
If you are not planning to fly with a club and just want to know the rules for a Class 1 flight, like maybe you want to fly on your own property but don’t want to get a waiver, then only the FAA rules apply, not the NFPA. If that’s the case, here are the answers to your questions:
Can I launch a G126 motor at a class 1 launch if I am high-power certified? Yes. You can fly motors over 80N average thrust if you are certified, and it will not affect whether the rocket is Class 1 or not. It might not be allowed at a club launch, but it does not affect whether a waiver is required.
Can I launch an H-class motor at a class 1 launch if the propellant weight is less than 125g? Yes. If you are certified or attempting a certification flight, you can fly an H motor under 125 grams propellant, and it is still a Class 1 flight. Again, it might not be allowed at club launches, but it would not require an FAA waiver.
Can I launch a G-class motor at a class 1 launch if the propellant weight is more than 62.5g? Yes, I think so. I don’t remember what the 62.5 gram limit pertains to, but I think it has to do with mailing or shipping motors? I’m pretty certain this does not have anything to do with Class 1.
I hope this answers your questions. Are you planning to go up to the edge of Class 1?