How To Stay Logged Into A Website And Have It Run In The Background

I have a website on which I will sometime receive incoming chat messages. But, it could be weeks between when I receive those messages. I would like to
(1) Stay logged into the webpage (I'd imagine that it times out after a while, although I am not sure of the frequency with which that happens).
(2) I don't want to see the browser window on the Windows 10 toolbar. I want it to just be running in the background and only have something pop up when I have an incoming chat message.

 

i would think that within the preferences of your account on that website, you should be able to configure the option to either receive a text or email when a chat is being initiated. if not, then contact the website owners and ask for this feature to be implemented.

 

Ok, that's a solution.
However, with incoming emails, most of the emails that I receive I do not want to be alerted about in real time. Instead, I prefer to let the emails sit and I can deal with them when I feel like it. But, there are other communications and emails that I do want to be alerted about instantaneously. I am not sure if email software (Outlook, etc) lets you do that. But, one workaround is to set up a forwarding rule from one email account to another and set up the 2nd account for desktop alerts (the forwarding rules would only be forwarding the emails that you want to be alerted about).