Some surge protectors work by "redirecting" surges to the equipment ground, which is pretty much the definition of a ground fault. In this case, it's definitely possible that surge suppressors could be tripping GFCI receptacles/breakers (given a large enough surge).
As for the AFCIs... When electrical contacts separate, they often create an arc between the contacts. This arc spans the gap until the contacts are too far apart, and the arc can no longer be sustained. It's possible (in theory) that the GFCI tripping could create a large enough arc, for long enough to cause the AFCI to trip. It's not likely that a properly functioning GFCI would do this, but if the GFCI was old and worn it could be opening the contacts slow enough that a detectable arc is being produced.
Determining if the GFCI tripping is causing the AFCI to trip, is a simple test. Simply press the test button on the GFCI. If the AFCI trips, you'll have figured out half the problem. If the ACFI does not trip, you could repeat the test using a receptacle tester with a GFCI test function.
![enter image description here](https://i.stack.imgur.com/RCjG1m.jpg)
If it's determined that the GFCI is causing the AFCI breaker to trip, replacing the GFCI receptacle may resolve that part of the issue.
Figuring out if the surge suppressor is tripping the GFCI, is a more difficult proposition. It will likely require you to monitor the supply voltage over some time, to identify and document possible surges.
You should never chain two GFCI circuits together. When you install a GFCI circuit, you should only ever chain standard outlets off of them. Multiple GFCI on the same circuit can cause each other to trip.
So yes, what you want to do is actually what you should do.
What you do need to do, however, is make sure you connect the new outlet to the LOAD terminals on the GFCI outlet. That will put the new outlet on GFCI protection.
You should also sticker the new outlets as GFCI protected.
![enter image description here](https://i.stack.imgur.com/XP5qQ.jpg)
See in the above how the duplex (standard) outlet is fed from the GFCI, NOT the main line.
And in case it's unclear- the 3 gfci in the pic are not chained - they are connected in parallel. Only the standard outlet is "chained" - or connected in serial.
Best Answer
Yes, it can be done. There is no problem having a GFI fed from another GFI. The only down side is troubleshooting. If something is causing a trip you have to isolate one first to see where the problem is.
My advice is to replace both receptacles with GFI and install a standard breaker. You get the same protection but more convenience. All for only a few dollars more.