I think I can help with the first, but, as you said, a native speaker would answer that question better.
I wish I could see what was around the phrase (before and after). What did you look at? If you're trying for a quick glance, that's una mirada.
I would also retain the que and use preterite, because a glance in and of itself is not something that is sustained. It's a quick action. Lo siento que miré. If you're taking a look at something, that's echar una mirada a ...
Disculpame por echar(te) una mirada (if were talking about a person.)
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