DKS is the combined remastered D2:ED together with the FoV add-on. Much of the content of the D2:ED section of DKS is the same as the original release of D2:ED, but there were also a lot of changes (improved graphics, better inventory from merchants, changes to opponent placement and loot, changes to enchantments so they are available earlier and have more of an effect, etc). If you don't want to try another build to play through the remastered D2:ED, though, you can also start a new level 35 character at the beginning of the add-on.
Your character/save from the original release can not be used or imported into DKS. The balancing changes mean you should be able to progress faster at the start of the game compared to D2:ED (you can be a level or two higher when you get access to the full Broken Valley, which was a difficulty jump in the original release if you tried going after the skeletons around the tower before the goblin areas and derelict tunnels).
I use the direction pad for mostly non-critical actions (summon creature, dragon stone, dragon morph stone and the healing spell), though have occasionally hit the wrong direction (a couple times in flying fortresses I transformed in dragon form rather than healing; once I was high enough I could transform back, once I wasn't and died).
For potions I pause the game (or use the low hit point auto-pause) and hit a shoulder button to bring up a menu to select one (or an item, or skill that doesn't need a target).
The left stick button is the same in DKS. IMNSHO it should only toggle combat mode if the stick is centred when pressed. Playing a ranger in DKS I sometimes had a bit of a problem in hectic combat early in the game trying to jump and dodge (wasn't an issue in D2:ED playing a warrior), but mid game rarely have an issue (I don't need to dodge as much, changed my technique and have probably just gotten used to the difference).