I really don't want this to seem like a criticism of you, but my mum just split up from an extremely jealous partner (fake facebook profiles just for stalking her, accusations of affairs with numerous friends, reading her texts repeatedly etc), so I can't help but see the other side of this.
None of the things you have described factually happening (a woman telling him to get in a car, being facebook friends with someone, being referred to as "babe", saying "thinking of you" in a text, even having lunch with a friend of the opposite sex) are in any way necessarily indicative of an affair, and none of them are behaviours I would be concerned about with my partner.
However, I trust my partner, and the real issue here is that you feel you can't trust him, and that - more seriously - he's compounding this by lying. What you really need to consider however is why he is lying. Is he lying because he actually has something to lie about, or if he is lying because he knows that what he considers innocent acts (such as being facebook friends with someone, or having lunch with a female friend, alone) would, if you knew about them, lead to arguments?
Lying is never the correct response, of course, and the latter is not an excuse for lying, but he could just have not told you about the lunch etc to avoid an argument - it doesn't necessarily mean he's having an affair.
Certain things you describe in your post suggest you might be prone to jealousy and insecurity in your relationship. I say this because I trust my partner in that respect, and because of this I would never in a million years look at his texts - I definitely wouldn't reply to one - or object to him having lunch with a female friend (alone, at her house, as he did recently - he didn't tell me about it till afterwards, but it didn't bother me at all, and importantly there was no lying involved). However, I've had partners in the past, when I was younger and more insecure and the relationship wasn't built on trust, and I definitely felt the temptation to look at their texts, so I know how you feel.
I honestly don't mean this as a criticism, but there's clearly trust issues between you and your partner, and it's really important to work out if these come from your insecurity, his untrustworthiness, or a combination of both. Do you have a mutual friend you could speak to about it? That might help. I'm sorry I can't offer more advice. :(