Yes, i understand she is 2 but its not a matter of her "giving in" or having control over you, etc... She cannot fall asleep and screams bloody murder because she is extremely overtired and she cannot relax enough to fall asleep. You need to help her out and this is where the longer wind down helps (well it helps me/us). Start nap routine at 11am so that she is in bed by 11.30 and if she is fighting you, at least you have some time before she is exhausted and cannot fall asleep at all.Honestly, the only way to get rid of overtiredness is an early bedtime. The more sleep she gets at the front end of the night (which consists of more deep sleep - restorative sleep), the better rested she will be in the morning. She might even sleep in past 5am. It will take a few days to catch up but its always worked for us.
If you feel that she is teething, then try giving her some pain relief.
Again, I know this is really hard. I was 9 months pregnant and about to pop J out when Isabella started giving me h*ll at nap and bedtimes. She was going through the 2-1 transition and I didn't know how to handle it. Then when J & I came back from the hospital, she refused napping alltogether and she was only 13.5months. Extremely overtired, cranky, clingy all day long and we had horribel NW and 5am starts. Take it one step at a time. If you have to, put a matress in her room on the floor just until she is sleeping better and then start sleep training so that she can sleep without you in the room....
Being scared of the vacuum cleaner, lawn mower, etc... is normal. Isabella went through a phase of being scared of everything that makes noises and its not until I started making up stories that she settled down. She even went through a phase of being scared of the oven and that thing hardly makes a noise and is out of her reach to touch....
Let me know how it goes
Layla