I don't have a history of that sort of behaviour. So I got to go on history of how I behave as opposed to my fear that I have suddenly turned into an uncontrollable terrible person.
Past behavior is a pretty good indicator of future behavior, so the fact that you haven't behaved like this in the past (bear in mind, though, that terms like "stalk" are extremely loaded words and most stalkers would not define any of their past behavior as "stalkerish" - again, we've known one another for a little while now so please do not take this to mean I think you are a stalker, lol) - and I only know you from what you post online, but you don't seem "stalkerish" to me. Those vibes are generally pretty glaring. The fact that you've owned up to it here, in the terms that you have, also suggests you do not fit this description.
You (seem to) have a fairly ordinary level of transference with your therapist for the level of treatment you engage in, the modalities you use, and the issues you're treating. You also demonstrate quite good adherence to normalized social boundaries and rules in your interactions on this website.
In Canada at least all the therapists I've ever known, email is not at all a common method of between-session communication (in fact Kayla is the first therapist I have ever sent emails to, beyond the scope of scheduling) - has your therapist just made it clear that
you can email her and that she simply won't respond until your session time, or was it communicated like "ZERO emails outside of session EVER" or whatever?
That would make a difference. My therapist doesn't do any kind of therapy over email, the only reason I email her is to send her journal entries or media relevant to the upcoming session. A couple of times I emailed her when I was psychotic, but even in a deeply dysregulated state, I did not expect her to reply. (She was nice enough to do so on both occasions, with brief answers.) But what I mean is, her boundaries are reasonable and normal, whereas something stricter than that seems unnecessarily... hostile? I guess? Toward clients? Especially clients with childhood trauma.
In a perfect world I would recommend being honest with her because it would certainly lead you to explore how your transference at times delves into unhealthy coping mechanisms (such as searching for your therapist's personal information when upset) and if your therapist were experienced with transference she would be able to get a lot of meat out of that session, and I am sure it would do a lot to solidify the therapeutic alliance you both have.
But again, the email boundary is weird to me, and if she's that harsh in one area, there is the potential for her to be even harsher when it comes to her data. Unlike others on this thread I don't require much of a therapeutic alliance to actually "get down to business," so I suppose there may be a bit of my own inability to fully understand the emotional component going on here.
My rationale is, if it were me, and I knew I'd be irreparably harmed if my therapist terminated me for a boundary violation, I could easily gain benefit from our sessions while electing to keep that to myself. But then again, the fact that I could gain benefit either way, is probably indicative of my lack of overall attachment to my therapist. Whereas your issues stem solely from your over-attachment. Still, I do encourage you to put your best interests first. Whether that is telling her the truth or omitting it.