The variations in 105 are either generation/cosmetics, or the cage length ie. short or medium (I don' think there's a long, but maybe), or possibly a special version of mounting system.
So, if your bike has a normal gear hanger built in then the standard RD is right. You still have to decide on cage length, based on the gear range. I don't memorize Shimano specs, but a typical road setup with 39/53 and 13-26 needs 37 teeth (14+13) of take up capacity and is the limit for short cage. More take up would want the medium cage.
And yes it should work with the existing Shimano controls. You don'[t need to change the front, since the front and rear derailleurs don't speak to each other anyway. As far as whether you should do everything, it's up to you, but I tend to be wary of trying to radically upgrade a cheap bike, and (without knowing) suggest keeping cheap bikes cheap ad plan B bikes and buying better if you want better. However, upgrading part by part as parts need replacement is a different story and is never a bad move (within reason).
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