Can I start off by saying that I discovered my husband was cheating on me last year so my judgement is somewhat coloured by that. Whereas before I trusted now I doubt so please accept that my view is partly based on my own negative experience.
As two separate instances, your DP screwing up the letter and this woman asking if he is okay are probably okay but together they would concern me. Unless he dislikes the woman immensely I don't see any reason to screw up the letter, it is strange behaviour.
I know the meaning behind the written word can be misinterpreted but why would she ask if he is okay as a direct question if they hardly knew each other. Surely it would have been said as a statement, as a hope everything is okay with you, without the question mark.
My hunch, and it is only a hunch it could be entirely innocent is that there may have been something going on, maybe as an emotional affair or even just a mutual attraction.
I don't know if your DP's changing job was sudden or unexpected but if it was that would lend credence to that point of view.
I am speculating that whatever the nature of their relationship your DP decided it was best that they didn't work together anymore before it developed any further and changed jobs. There are still feelings there on her part hence the asking if he is okay? Because your DP didn't want to be tempted he is not friends with her on Facebook because he has taken steps to not have this woman involved in his life. Possibly this woman is purely Facebook friends with some of your DP's friends because she took an interest in his life and they responded to her requests because she said she was a friend of your DP. It is possible she doesn't know them in real life.
Alternatively this woman liked your DP and he wasn't remotely interested but felt uncomfortable with her level of interest and thus changed jobs, isn't Facebook friends with her and that would also explain why he screwed up her letter.
Is there any possibility that she was an ex-girlfriend of his from before you got together. He could have thought you would be uncomfortable with them working together and kept his distance and could explain the letter and how she knows some of his friends.
Please speak to him and I hope he can reassure you that you have nothing to worry about. All of my thoughts could well be way of the mark but may give you some insight into what you would like clarified.
Congratulations on your pregnancy and I hope all turns out to be okay.