Let's say I want to play "Master of Puppets" by Metallica but I like the heavy sound of D standard. Can I play the riffs in that tuning using the same chord fingerings as in E standard to get the heavier tone? Another example would be playing "Last Resort" by Papa Roach in drop B instead of drop D. This would lower the key of the riffs correct? The songs cannot be "broken" this way as long as the relationship of the strings to each other stays the same right?
You are correct that the songs will not be "broken" in the sense that they won't sound totally different. Every single note will sound lower, but the relationships between the notes will be preserved - because every note will be lowered by the same amount.
As you note, that will make the overall song sound a little different. It will make each song sound "deeper" and possibly thicker and other interesting metal-sounding words. Mostly that's a good thing. Partly that is caused by the actual pitches involved and partly it is caused by how the sound of the guitar, amp, and distortion change when you play lower notes versus higher notes.
In any case, it can be that some people won't like songs played lower. It will sound different, and some people might not like that as much. Also, some songs might sound worse to most people if played in a lower tuning. Most people might prefer "Here Comes the Sun" by the Beatles in its recorded key, and not as much in a lower one. So some songs might be a little "broken" for some people by being played in a lower key.
If you you like it better in a lower key, then play on! Not only are you playing partly for your own enjoyment (I assume), but also anyone listening will be shown your interpretation of the song, which to me is a lot more meaningful than just re-creating the song perfectly exactly the way it is on the album.