Can Slifer be summoned using Red-Eyes Darkness Metal Dragon's effect?
Okay, I think I've worked out what's going on here.  The link to Original Slifer that you kindly provided for me shows the original, illegal version of the card.  There is a second link, from that page, to the legal Slifer version.  You will see that the second Slifer does not have any ruling declaring the card to be of Dragon Type.

Why does the illegal Slifer have a ruling indicating that it is a dragon?  Well, it took a little bit of detective work but I quote from that page:


  This card's written lore in the anime and manga was never shown fully
  or directly; the lore listed here is a reconstruction based on how it
  was used and may not be accurate or complete.


What is this talking about?  Well, at some point in the anime an interaction between Slifer and other "cards" occurred that would not have been able to happen if Slifer had not been of Dragon type (I believe I actually found what this was earlier today, but have lost it again: I'll edit it in if I track it down).  Seemingly, then, in the anime at least Slifer is of "Divine-Dragon" type.

In tournaments I would guess that Slifer is categorically only a Divine-Beast, and not a dragon, and does not combo with Red-Eyes Darkness Metal Dragon.  You say you're not especially interested in tournament restrictions and the latest rulings though - so if you wanted to make a house rule that yes, of course "Slifer the Sky Dragon" is a dragon, then there's nothing stopping you.  That interpretation was good enough for the anime so there's no reason it shouldn't be good enough for you.  If you go down that route I would treat Slifer as having "Divine-Dragon" on its type line for all purposes, so it would still be a Dragon in the graveyard too.  Hope that helps!