So when will this happen
This isn't such a trivial thing to do. You'll find even when hosting your own server there's a limit to how far you can scroll back. In terms of doing this in the web console, dumping hundreds of thousands of lines into your browser at once when you open your console if your server's been running for a while isn't going to play so well with your browser.
I can't say it's a priority to look at for the moment but it's something I'll definitely keep in mind. It may just end up being a button to download the current console.log or something.
Is there a way to stop messages from showing up under the server chat? In my CRP if a player enters 1 to withdraw from a bank it prints "Player: 1" in the server chat tab.
Not at the moment unless you want to edit the System_RTBHosting add-on yourself. Due to the packaging order it'll pretty much catch anything people say regardless of whether underlying packages decide not to call the parent.