Can i travel with j1 while petition is being filed for h1b

[Waiting9]

I am a university professor, currently in the US on J1 Visa. I am starting a faculty position at another university and this week they started the petition for the H1B visa. The application will be submitted with premium porcessing to expedite the prcoessing and get the visa before September 1st. My current J1 visa will end on June 30th. I have two questions:
1. I have travel plans to Mexico next week for 10 days. Is it safe to travel while the university is preparing the H1B documents for filing?
2. Since J1 visa will end on June 30th, if H1B would still be pending, what is my status in the US?

Thanks.