I'm not an electronics tech, but I do sometimes repair audio equipment for cheap thrills (and the babes, lots and lots of babes!)
Issue 1, speaker compatibility:
The wattage rating on a speaker is essentially how much heat the speaker can handle (or dissipate) before blowing something up - typically this means melting the varnish on the wires in the voice coil, thus shorting the coil, dropping the impedance, spiking heat and 2 seconds later, letting out the magic smoke.
A massive 700 watt amplifier played at modest volume levels is only putting out a watt, maybe two, to the speakers, so you could run a cheap little 20 watt bookshelf speaker and be fine.
You only run into trouble when you send more wattage to the speakers than they can dissipate. For 120w speakers, that's quite a lot.
Issue 2: speakers damaging amplifiers:
The only time I've seen a speaker f up an amp is when the speaker's impedance is too low for the amp. 4 ohms is a heavy lift for most amps - even those that are rated for it, and some speakers that are *rated* at 4 ohms may impede only 1 ohm or less at certain frequencies. I once built a huge speaker with double 12" woofers, 4 ohm dome mids and ribbon tweeters, and the crossover setup that gave the best sound also gave about .5 ohms from 1Khz to about 2.5Khz. The sound was beautiful, creamy, balanced, crisp, clean... and it would shut down the amp - any amp - after two or three minutes at volume. The low impedance was just too much and the protection circuit kicked in, shut down the amp till it cooled off again.
So assuming your amp has a protection circuit - and I'm sure it does - you are not hurting the amp with reasonable use.
Issue 3: intermittent pop
I've seen this before 3 or 4x and it comes from a bad connection. Now, that could be a bad connection in a cable, or it could be a cold solder joint on a circuit board. But something in the signal path is losing and regaining connection. Try wiggling or twisting your RCA cables and connectors. If you can induce a pop, then you found your bad problem. Replace the cable and live happily ever after. If you can't recreate the pop, you may need to look deeper. Finding bad (especially intermittent) connections on the boards is a big job for mere mortals and may be best left to a pro.
Hope this helps, but please remember, I'm just a hobbyist and my advice is worth exactly what you paid for it
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Please let us know how you solve the issues