Have a summer internship offer, but still want to network elsewhere for full-time just in case?

Hi all,

I am looking for some advice regarding a situation I am in.

I recently got a BB internship for equity research (my dream internship) and I am excited for it. At the same time, I still want to expand my network at other firms just in case full-time recruiting is necessary. I have several questions and was wondering if other people who were in similar situations can provide some insight:

1) Is it okay to email professionals at other firms while I have this upcoming internship? I won't be asking for a job, it is just so I can establish a connection with them and learn more about what they do.

2) Is it okay if I mention that I am an incoming intern at XYZ firm and found their background interesting/ would like to learn more about themselves? Or should I not mention my firm at all? I feel that by mentioning it I increase the risk of something bad happening like a professional at the firm I am emailing reaching out to his/her buddy at the firm I am interning for this summer.

Any advice would be greatly appreciated.

Thank you!