What are your go to questions to ask to "tactical stakeholders" and “Good To Know” connections during 1-on-1?

I'm going to approach this in a slightly different way.

First, there's probably a better frame than looking at the person you're speaking to as "tactical" -- to preface, I understand the initial context it's presented in, which makes total sense, but anchoring down on it will lead to the wrong mindset going in.

A lightweight framework that's worked well for me to break the ice with people I don't know very well both internally and externally (e.g. think intro call from cold outreach) is to go in with a genuine interest in discovering what problems they are trying to solve and be part of a joint effort to pave the way for a solution.

The conversation would naturally flow from there since it shifts the attention and thereby the pressure from finding some way to connect on a personal basis to two people hacking away at an interesting problem and letting serendipity take over. The things you'd usually cover in a more rigid manner such as what skills you bring, what you're past experiences are and how you like to work are shown rather than talked about through this process.

I like Casey's answer a lot. I also think just stating your intention is very powerful and disarming.

I just want to chat with you: understand what you're working on and see where I could help.

People are generally happily surprised to hear that. You're not requesting anything from them or adding a task to their backlog.

See also Boz's career cold-start algorithm:

The first step is to find someone on the team and ask for 30 minutes with them. In that meeting you have a simple agenda:

• For the first 25 minutes: ask them to tell you everything they think you should know. Take copious notes. Only stop them to ask about things you don’t understand. Always stop them to ask about things you don’t understand.
• For the next 3 minutes: ask about the biggest challenges the team has right now.
• In the final 2 minutes: ask who else you should talk to. Write down every name they give you.