I can help with the battery question.
I have that phone, as well as a few friends, and the battery is notorious for being a energy sucker. This model of the LG phone just is packaged like that. Android program itself has a lot of background apps that can be on, without you realizing, which is why it is important to always check the Task Manager. If any are running, you can turn them off and modify some of the usage in your settings.
Regarding the wi fi...
What does it do when you try to turn on the wi fi? How do you attempt to turn it on? Do you go through settings or do you use the drop menu from the home screen? Does it stay lit up or just randomly turns off? Has it never come on?