Can I export temperature and barometer data to the Phone and PC?

You could try analyzing some of the FIT files on the watch, using the FIT SDK. Have a look for example at the FitCSVTool at https://developer.garmin.com/fit/fitcsvtool/ I do not know though, where the data is exactly stored. I see some temperatures for example in the files in the folder //GARMIN/Summary, but am not sure whether they are exactly what you are looking for. Some of the data may be also within the Monitoring folder. The problem is that the CSV tool converts many fields as Unknown, so you have to try to find out what they are by other means (testing, comparing values, using other FIT file tools,...)

Thank you! Sounds like I need a wired connection every time

Well, another possibility is writing a Connect IQ app, for example streaming the data to a server that you create. Or perhaps there is already such app. Try to check out the app store

I1 doesn't support CIQ.

... ah, I see now you have the old Instinct. In that case, the CIQ app won't work for you, since the 1st generation of Instinct does not support the CIQ technology. Time to upgrade to GI 2 - I did, and could not be happier