Maybe you'll be able to check your bags through, maybe not. I suspect the latter. All terminal have left luggage, not all have (public) showers.

In terms of checking bags at LHR, if that's what you have to do - it will be at T4. I'd suggest clear immigration/customs and head straight over to T4. If you can check in do (though I think you'll be told you're too early), otherwise head to the ground floor where left luggage is, and right next to that are the public showers. Then you can do your thing and head back in time to check in.