Follow your instincts, even if they’re incorrect, you need to address them. If you want to know, ask (don’t demand). Explain how you feel clearly and calmly.It’s entirely your decision whether or not you believe him.
In similar situations, I’ve always been very upfront and open about my relationships with my closest friends. They are all female which can cause a lot of strain for my s/o if not properly addressed. I certainly do not share every communication between us with her nor do I feel it is appropriate to do so, but I always address any concerns my s/o has immediately and honestly. Trust and communication are the foundation of any relationship.
It wasn’t necessarily wrong of you to ask to see the text by the way, it’s dependent on how you asked within the context of your relationship. But, right or wrong, it can be perceived as intrusive or indicative of a lack of trust on your part. She may be nothing more than he presents, a friend, but clearly you both need to discuss it if it is continuing to bother you.
As to whether you should “know” the really good friend. It would be strange for you not to know of them, but not necessarily know them beyond passing acknowledgement. Of my three closest friends, my wife “knows” them all but she really only knows, to any depth, one of them, though she’s getting to know a second slowly, as my friend has been around more recently.