Barbarians (and Assassins, in the expansion) are the only classes who can dual wield.
Barbarians can dual wield any one-handed melee weapons - one-handed swords, one-handed axes, one-handed hammers. They can also dual wield throwing weapons like javelins, but without the Double Throw skill they can't throw from both hands. As a final point, they can also wield a two-handed sword in a single hand in order to dual-wield. This does reduce the damage, and the power of the weapon for both single-handed and two-handed version is listed on hovering. Assassins can only dual wield their claw weapons.
When you perform normal attacks, you will alternate between the weapons in your hands. The attack speed is calculated for each arm swing, so different weapon speeds will result in different attack speeds. However, the speed is generally faster than if you just swung either of the weapons twice. Clicking faster has no effect, and the damage is simply that of each weapon hitting. And of course, dual wielding is necessary to use a number of combat skills like Double Swing, Double Throw, and Frenzy for the Barbarian, and some martial arts skills for the Assassin.
Your right hand strikes first, but in most scenarios it's fairly moot whether you keep your weapons on one hand or the other.
Being a fairly avid D2 player, I can tell you that it works in roughly the same way. Just like in D3, as you go up in difficulties, creatures become much stronger. In D2, many of them gain immunity to specific elements, and the likelihood that you would encounter a unique (yellow/gold monsters) or pack of champions (blue monsters) is increased. Additionally, uniques and champions have more modifiers when you encounter them (similar to in D3).
In much the same way, the Acts also scale in difficulty. In fact, Blizzard did a very good job of making Act 5 harder than Act 4 of the same difficulty, but not so much that it was much harder than Act 1 of the next. A lot of this comes from the fact that as you increase in difficulty levels, a lot of your stats are reduced. For example, in Nightmare, all players suffer a -50% reduction to all resistances. Additionally, life steal effects are cut to a quarter. When you go to Hell, the player starts with -100% resistance (yes, it was completely possible to have all negative resistances, which would, as you'd expect, amplify damage taken from that source), and life steal effects were cut to 1/8.
Another big spike in the difficulty was also mentioned above: the more common appearance of uniques and champions. In Normal, it is unlikely that you will find more than 1 or 2 packs of uniques or champions in any given area (for example, the stony field), excluding super uniques which always appear in the area (like Rakinishu, Blood Raven, Bishibosh, etc). However, in Nightmare, it is not unlikely that you will simply encounter 2 packs of uniques/champions at once. Then you might have to deal with something like "Cold enchanted, extra fast" at the same time as dealing with "Lightning enchanted, multiple shot". That, along with natural monster immunities (for example, the entire Fallen tribe becomes completely immune to fire damage in Hell mode) makes Hell mode much more challenging than even Act 5 of Nightmare.
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You probably need to "mod" Diablo 2. This includes getting a MPQ extractor, getting some silent .WAV files and putting them back into the archive. This will work on Battle.net too (if I recall correctly), as sound effects and music are client-side.
You can get started by visiting the Phrozen Keep, the site for Diablo II modding. You're looking for