Find the gap first because the second that turn signal goes on people speed up.
Dependent on situation, but generally speaking, I look for the gap first, then signal, THEN change lanes. So, kind of a hybrid between your top 2 choices (did not vote).
What ever method you do. STAY OUT OF THE LEFT LANE.
The left lane is the "PASSING" lane. Not a fast lane, or a travel lane. Be there long enough to pass the car on your right, then you yourself need to move right. PERIOD!
Yes, many drivers see the turn signal as a sign of weakness and will deny you your lane change on that basis.
They're the same ones who view a yellow light as a "speed up" light.
Other. Depends on traffic conditions.
Normally, I will look for my gap, then signal, then move (but still, I turn on my signal before I start changing lanes). However if traffic is heavy and there are no gaps, I will turn on my signal to indicate that I intend to change lanes as soon as I safely can. It lets drivers in the other lane know I want to come over, so hopefully they'll let me in.
My husband never uses his signal until he's actually changing lanes (as in, starting that sideways movement), and gets really ticked when nobody will let him over in heavy traffic (although I'm not sure how they're supposed to know he wants over in the first place.) To him, the signal is to show you ARE changing, not that you intend to change.
When we lived in Arizona, we called using your signal, "giving information to the enemy."
My Dad taught, "Signal your intention, not your action," so I tend to always signal first, but mindful of the gap or lack there of. I hate changing lanes, so I tend to stay in the middle or even right lane, even if the car in front of me is slow. If it's heavy traffic, then DH is driving.