"Went" and "asked" are each simple past tense, and they set the context for the rest of the sentence, which must be from the point of view of when the "went" or "asked" happened.
1b) He went through her phone and found out she was lying to him all the time.
So, from that perspective, "was lying" indicates something that is happening in the current time; she is lying right now.
But the phone shows nothing about what is happening now, only in the past.
These two tenses don't get along here, and this sentence thus sounds awkward.
1a) He went through her phone and found out she'd been lying to him all the time.
She had been lying in the past, not now, and that fits correctly with the given perspective.
The second example is almost the same, but not quite.
"I had been missing" is correct for the same reason as above, but "I was missing" could also be correct, depending upon context:
a) … made me realize what I'd been missing. So we arranged to watch it next week.
This is like the first example.
The "*had been missing" is in the relative past, while the "missed" and "arranged are in the relative present.
They fit together correctly.
b) … made me realize what I was missing. So we watched the film right away.
In case "b" though, "I was missing" is in the speaker's present.
But that's okay here, because the "we watched is in the same tense as the "made me" and happens in the same time sequence.
One trick you can do is to reword the sentence into the perspective of the initial context:
1a) He goes through her phone and finds out she has been lying to him all the time.
1b) He goes through her phone and finds out she is lying to him all the time.
[The "is" is present tense, but he doesn't know she is lying right now. so this is wrong.]
2a) My friend asks me to watch that film and makes me realize what I have been missing.
2b) My friend asks me to watch that film and makes me realize what I am missing, so we watch it right away.