I'll try answering this question (my first attempt at this site). I transitioned years ago, and received messages from unexpected people reaching out.
To use her correct name, or not?
Surely it's clear that deliberate deadnaming is not a good idea.
From my point of view, if someone uses my name and pronouns, I automatically feel significantly more at ease (especially when I know they know about my past, and they're choosing to address me correctly despite being capable of doing otherwise), so I encourage that. There's probably 20 different non-stalker ways of accessing this information. If it comes up, just say:
I thought it was better to use your correct name, so I looked it up.
But really, transgender people are aware that word gets around. Spreading around someone's correct name to prevent deadnaming is perhaps a benevolent kind of gossip. I don't ask how people know my name.
The most important thing is to interact with her basically as you would any other woman in the same circumstances, with some allowances.
Some possible pitfalls...
What made me feel uncomfortable, which hasn't been addressed by the previous answers, are people reaching out to score transgender brownie points. I got the feeling they only wanted to know me so they can subsequently argue that "I have a transgender friend" or "I'm a transgender ally". (Either that or to gather dirt.)
Also, don't be too surprised if they're not that interested in being long-distance friends with some random person from the past. There's a huge number of reasons for this: anxiety, being introverted, being satisfied (or feeling safe) with their current group of friends, busy doing something else with their life, ten other people have done the same thing you're doing and they're sick of it, and so on.
I also wouldn't bring up uncomfortable discussion points, like sensitive family issues. For me people would bring up these topics, and didn't understand that I already have other people to discuss them with, and that I don't want to spend my entire time reliving painful memories, I want to have fun with my friends. In fact, many transgender people don't like being transgender, possibly considering it the worst aspect of their existence. I don't want random people from the past contacting me out of the blue:
Hey you're transgender, and therefore your family hates you! Want to discuss it?
No. No I don't. Others might. I don't.
I strongly recommend avoid proposing "solutions", especially the quick-fix, rewrite-your-whole-life kind of solutions. This interference is highly unwanted and massively stressful. I've had people propose that I spend thousands of dollars on lawyers, move to another country, quit my job, take various medications, and other things involving I won't even mention. They don't understand that poor mental health entails I cannot even do the things they're proposing. Then they get grumpy at me for not listening to them.
Of the random people who reached out to me, most (maybe all?) subsequently forgot me immediately after. It leaves me feeling that they contacted me as a perceived "social requirement", not because they actually care. This in turn feels degrading: it feels like they assume I am so desperate, that I'm willing to accept a fake or manufactured friendship.