These are parenthetical commas - ie, they function like parentheses (aka brackets or braces) within the sentence. So to see if they're right you could switch them for actual parentheses. Eg (I've added 'an' before 'infinite' as I think it needs it - this isn't stricly relevant to the question but it will clarify things).
I began installing a new system (or ROM) on my phone every weekend and (in the process) broke the poor phone an infinite amount of times.
The parts in parentheses add more information, in addition to the main thing you're trying to communicate. If you took them out, would the sentence make sense?
I began installing a new system on my phone every weekend and broke the poor phone an infinite amount of times.
Yep, that's fine - it works without the parentheses, so the parentheses are correct, and therefore the commas are correct.
In total, my victim phone got broken 26 times, and, every single time, I found a way to revive it
could become
In total, my victim phone got broken 26 times, and (every single time) I found a way to revive it.
without the parentheses:
In total, my victim phone got broken 26 times, and I found a way to revive it.
Hmm, this feels wrong. It now sounds like "I found a way to revive it" happened only once, not 26 times. This alerts us to that fact that "every single time" can't be removed from the sentence in this way. Which, in turn, means we need to revisit the commas. I'd rewrite the original as:
In total, my victim phone got broken 26 times, and every single time I found a way to revive it.
This is better, I think.
Finally:
English opened the doors of an entirely new world—one brimming with knowledge and values not confined to the limits of my country—for me to explore, and, as I began to explore, my childhood fervor for playing games paved way for a much implacable thirst for learning.
We've got a new element here - the em dash, often referred to as simply "dash", or as a "hyphen" (it's technically different to a hyphen (it's bigger, for example) but this isn't terribly important, I think. Em dashes are harder to do with a computer keyboard and so people often use hyphens instead). It performs a similar function to the parentheses, or parenthetical commas, of inserting a subclause into the middle of a sentence.
Keeping the em dashes but parenthesising the rest of it, we could rewrite it as
English opened the doors of an entirely new world — one brimming with knowledge and values not confined to the limits of my country — for me to explore, and (as I began to explore) my childhood fervor for playing games paved the way for a much implacable thirst for learning.
If we took the parenthesised section out, would the sentence work?
English opened the doors of an entirely new world — one brimming with knowledge and values not confined to the limits of my country — for me to explore, and my childhood fervor for playing games paved the way for a much implacable thirst for learning.
Yes, this seems fine, which tells us that the parentheses work, and therefore that the parenthetical commas work.
So, I think in summary that examples 1 & 3 are fine, but 2 isn't quite right.
