1. Is it OK to resize the dialog to accommodate changed content?
Yes it is nowadays, providing it doesn't cause the window to scroll offscreen. There are lots of ways to do validation notices, but the approach you've chosen (notify below the field, red, left-aligned) is a very well tested and classic approach so it's a safe choice.
The main downside to resizing the window is the momentary disorientation the user feels when the messages appear and the form suddenly shifts. To ameliorate this, I'd suggest using a slide-down or other animation (e.g. this) to provide a smooth transition so the user understands visually what is going on.
2. What other options are there? Ideally we would like to keep the window compact - without large unused/blank areas.
There are other ways to getting this done without creating a lot of unnecessary initial white spaces, as you say (e.g. floating messages) but I think your instinct around the slide-down messages should work just fine unless you really want to get fancy.
If you want to prevent the modal dialog from being resized, I would suggest adding extra space between the form and the Join/Cancel buttons as that is the conventional way to do this. i.e. the buttons should be aligned to the bottom of the dialog, and there can be open vertical space between the Remember me checkbox and the Join/Cancel buttons.
BTW, it's not great UX to provide messages like 'Invalid Meeting ID'. To the user, this feels like you are scolding them without providing them guidance around what to do. Rephrasing it as 'Please enter a valid meeting ID' is better UX because the user is being directed on how to fix a problem rather than just being alerted that there is a problem.
