I think the best way would be to have a look at BrowserStack website, they actually answer all of your questions.
However, to point you a bit to the right direction, as of May 2021, BrowserStack offers these products:
Out of these, you will likely be interested in these 3:
because they let you test in desktop, tablet, and mobile browsers on various devices and operating systems. A quick look is this screen, you can see there're lots of options:
Live product is with no automation, probably not what you're looking for right now. Automate lets you execute various scripts (you can choose here, Cypress is also supported). Percy is a visual testing tool, it basically takes screenshots of your web app and compares them against some baseline you feed to it.
Now on to you questions:
Do we have 'record and replay' kind of a setting with BrowserStack
BrowserStack is not a tool that will generate any code, it only runs code that you're provided on devices and in browsers you specified. So if you choose a tool like TestCafe Studio, Cypress Studio, or similar, then you can have record & replay feature, but that's something you prepare locally and only then send it over to BrowserStack to execute that code.
We are also developing the UI automation using Behave + Python and so far we have successfully automated our smoke test suite [Regression is in progress], can we also use the existing selenium scripts with BrowserStack [ I know the answer is YES but this will only cover different browsers and not mobile devices, I guess]
Mobile browsers and devices typically display a page differently than desktop browsers. What usually happens is some elements are hidden in mobile browsers, or some other elements get used. If you've created a script that is tied to a desktop browser, it might be a problem because some selectors will just fail, therefore your whole script will fail. This of course depends on a particular app (which we'r not familiar here), so there's still a chance everything will work smoothly on all platforms, but I'd guess it won't and you'll have to spend time adjusting your scripts.