I agree unreservedly with Math here. However, I think there needs to be some honesty on both sides. At the moment, understandably Daisy you are keeping some cards back; understandable because you suspect this man is playing games with you and will not be honest.
If you believe as you say that he was on those sex sites, it doesn't matter for the moment what you believe his motives were, the fact is he denied even accessing them. You need to tell him that you believe that was a lie. Then perhaps if he admits that lie, you can find out why he was lying and what were his true motives for going on those sites.
If he says it was curiosity, then you both need to discuss where that curiosity came from and whether it is related to an aspect of his sexuality (sex with strangers) that cannot be met in your relationship. If it's that, he either needs to promise he can quell that desire or tell you honestly that he cannot. That gives you your choices back, to trust or walk away, doesn't it?
Similarly, the text issue has gone far beyond the acts themselves. You need to tell him you've found them and you don't need to lie about how or why. If you've always believed he was lying to you about those sex sites, you understandably didn't trust him. If however you genuinely found them by accident, then say so.
You need to find out why he was sending them and FWIW, I think the content of them does allude to fantasies that need exploring, especially as he has previously admitted desires for you to engage in a threesome.
What's more serious though is that he has admitted to setting traps and then refusing to discuss that. That needs unpicking too and it is wholly unreasonable to refuse to discuss unacceptable behaviour. Setting traps in a relationship is harmful, whoever's doing it. If this "trap" business was in fact a lie to cover up what motivated the text content, then he needs to say so. It wouldn't be the first time an instant lie actually showed a person in a worse light than the truth.
I can see how an instant lie to the extended family and friends about the postponed wedding might have seemed attractive and stopped questions about what are after all, personal matters, but it would have been better with hindsight, as Math suggested, if a nebulous "we've got some personal issues to iron out" had been offered. Hence, I wouldn't compound any lies to your future MIL Daisy - I would be honest, without giving specific details and ask her to respect your privacy as a couple.