Hi, welcome and thank you for your question. My name is ***** ***** I am glad to assist you today.
I am very sorry about your issues. If you are using Gmail through a web browser, yes, it is using a private and secure connection over the internet. You can verify this on your web browser where you can see a locked key icon or an HTTPS on the address bar. The "S" there means secure.
Please check this screenshot of my Gmail accessed through Chrome:
Now, regarding the prompts that you are getting, they are imposed by Gmail for your protection against common hackers nowadays. If someone is trying to guess your password ***** failed to login several times, Gmail will notify you through your phone. So it is very important that you keep your phone number UPDATED on your Gmail account.
I hope that helped.
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