1) When girls get together, especially if we're really stressed out about something, our collective maturity tends to drop like a rock. You should see me and my two best friends... we go from sane 25- to 30-year-olds to giggling 5-year-olds in the space of about 10 seconds. It helps us stay mentally stable, don't whine about it. If you can't stand watching it, DON'T BE THERE. Simple.
2) Guys tend to forget that girls get majorly insecure. Most girls I know talk to random guys and think nothing about it because we don't care about the guy and aren't thinking about jumping his bones. However, girls tend to forget that guys can also talk to random girls without thinking about how they look naked. I'm pretty sure that's the problem with yours- she remembers you two used to be friends and now you're more, so she may be scared about another girl doing the same thing and replacing her. Ask her if this is the case up front, and then tell her all the reasons you don't want anyone but her if she says yes. It might help.
3) If you're sick of her lying, tell her that the next time you catch her in a lie you're gone, because you can't live without an honest partner. MEAN IT. If you catch her, get your butt out. No one deserves someone who lies to their face, even if that person is only lying because they feel they'll lose a relationship if they don't pretend to be someone else.
Remember, there are times when loving someone, no matter how much, isn't the right thing for you. If you still can't get through to her, let it go and move on. Take care of your heart, even if she won't. Good luck!