That is less important than you think. What is important is that you have experience that shows you can function in an academic research setting (the gap between physics research and engineering research is often smaller than you think) and it should lead to a letter of reference from said professor that is stronger than what you would otherwise have.
In general, yes. You circumstances may vary a little bit such that a second bachelor’s degree makes sense (and it taking only one extra year is certainly a bonus), but in a general sense, it is better to get an MS than a second BS. Do you have any sense of which engineering field you would like to study?
There is no magic GPA to get admitted. I was admitted to a “top ten” program or two with a 3.3 overall GPA and something like a 3.4 or 3.5 GPA in my technical courses, a couple years of research experience, and GRE scores that didn’t disqualify me (since no GRE score alone will make you stand out). They definitely care more about your GPA in technical courses, and typically more about your last few years in school rather than how you did as a freshman. Basically, if your GPA is slightly below average but you show marked improvement in your last few years, you have a better shot. If your GPA has gotten continually worse, that hurts some.