My son got his first tooth at 4 months. It's uncommon but not impossible. However, if she's teething she'll also be showing other signs of it like drooling a bunch and chewing or rubbing her gums on things.
If she has an ear infection, the doctor should be able to tell right away. She'd also probably be pulling on her ears and crying harder when you put her down to sleep. The pressure of an ear infection hurts more when you're lying down.
Understand also if this is your first baby that sometimes they just get into moods and cry for no discernible reason. I'd look into all other things first, but remember that babies don't have any other way to communicate with you so sometimes they just cry and there's nothing you can do.
I have a lot of luck with telepathy with my baby. If you have either telepathy or empathy you can project yourself into her mind or her emotional state and usually get a feel for what's wrong. My son finds it soothing when I do that. Every baby is different.
Finally just remember that if you're doing everything you possibly can and she's still crying it doesn't mean you are a bad parent. Just keep trying things and eventually she'll stop fussing.