In typical freeform textplay it's relatively common to narrate basically everything except for the actions and reactions of other people's player characters. This applies more the longer and more spaced out the posts are - trading quick responses on chat doesn't exhibit this characteristic as strongly as longer, more elaborate responses or forum posts.
The exact boundaries of this need to be negotiated by the players (they're also often part of forum rules on roleplaying boards), which is one thing you could do. You can simply go "Hey, I noticed the last time that we're working from different assumptions on how to play our NPCs. I'd prefer if we did it in such and such way, how about you?" and try to establish a common ground.
As an alternative, if maybe your partner prefers to have more options in terms of posting you could negotiate that whenever someone introduces an important NPC they accompany their post with a meta-post, containing basic information about who they are and how they think and act. If you're leaning far towards the collaborative story-writing angle of text RPG, this may be the better solution for you, as it allows for longer and more elaborate posts without needing to consult the other player(s).
For the part about the interruption, my first approach would be to simply check if there wasn't a misunderstanding. They might well have read your last post in such a way that they thought your character had interrupted theirs - one's own writing often seems much more obvious and clear to oneself than to others. So simply go and ask "Hey, I noticed you wrote that X interrupted Y, but I didn't mean to imply anything like that in my post. I thought it went like this and this. Do you want to edit that?".
If they insist on playing it their way, either go along with it or take the misunderstanding into the play itself and let your character go "Huh? I didn't interrupt you, what are you even talking about?". This is a little confrontational and can lead to conflict between the players, but if you feel like they're genuinely encroaching on your character's agency you can try and make a point like this. Don't say I didn't warn you though. I suggest letting it go this time at least and using the second method only if it happens again and they refuse to acknowledge your own character's actions as written by you once more.