There is actually a place that converts Amiga classics into easy-to-use Windows executables that work on modern operating systems and allow you to easily customise a good deal about them, including resolutions, screen filters, controller support, save states, etc. using the UAE emulator, all for free. They actually have Speedball 1 and 2 converted into a single executable. But I'm not sure I should say any more or link them because I'm not sure how legal it is, especially when they've also done many other GOG titles that were on the Amiga (Cannon Fodder, Syndicate, Populous, etc.). They've even been getting some of the CD32 titles done recently, incorporating their CD soundtracks.
With news of Microsoft recently working out some deal to get a bunch of Amiga classics released for Windows and tablets, Windows phones, etc. I imagine they might not last that much longer if MS has anything to say about it anyway.
It's most likely not legal since the UAE emulator needs a kickstart rom to work, and those come with a fee. However, I can highly recommend Cloantos Amiga Forever package which also relies on UAE but makes it almost as easy to run old Amiga games as you describe it. Cloanto has licensed the Amiga Kickstarts so it's perfectly legal. The package comes bundled with some games but there's lots more to download from various sites (of course I'm only talking about legally free games. I doubt however that Bitmap Brother's games have been released as freeware so you would need to actually own the games. But other companies like Cinemaware and Team 17 have released some or all of their Amiga games for free. So, if you want the pure Amiga experience without actually buying an Amiga I think Cloanto's Amiga Forever is the way to go.
As for the MIcrosoft comment, it won't be them suing anybody, it will be the license holder Amiga Games Inc. which I mentioned in my earlier post. I think Amiga Games Inc has the rights to around 500 games at the moment and they seem determined to get them out on as many platforms as possible.