Does being right or left handed of any importance in a fight? This is probably broad but let’s keep it with weapons like swords, daggers, mace, clubs and the like??
Somewhat. When we’re just talking about using the weapon in a vacuum, it doesn’t matter that much. Similarly, when you’re talking about a duel, where both combatants have a sword, and nothing off hand, it’s not a huge deal.
Being a lefty becomes important in combat in some specific situations.
If you have an off hand tool, such as a parrying dagger, things can get nasty. Parrying works by redirecting the attack away from your body, usually by pushing the attack out. When someone parries with an item in their right hand, they’ll push the attack to their right (their opponent’s left.) Normally, with two right handed users with parrying tools in their off hands, this will allow them to parry each other’s weapon. However, when a left handed user parries a right handed opponent, they will redirect their foe’s weapon arm across, blocking any potential parry.
Worth noting that this goes both ways, and that a combatant facing a foe with the same dominant hand can choose to parry with their primary weapon to strike with their parrying tool, if it can be used that way. (Not all parrying tools can be used to stab your opponent. This is especially true with some varieties of swordbreakers.) Also, if you are parrying with your weapon, to shank someone with your off-hand, you’re going to close the distance.
None of this should really come as a surprise to an experienced soldier. Fighting a left-handed user isn’t as common, but it’s not unheard of.
Also, worth knowing that some duelists will specifically train to fight with their off-hand. It’s intended for showing off. A sort of, “I’m so good, I don’t even need to use my dominant hand.” I’d say it’s not a good idea, except it does have applications, such as if you cannot use your dominant hand for whatever reason. So, it’s probably more accurate to say that it’s not practiced with a practical goal in mind, however it can be useful in rare situations.
Most real castle architecture was designed to favor a right handed, defender. I’m going to use a specific example here, but many spiral stairways are designed so a defender, fighting from an elevated position, will have more room on their right side, giving them more options from which to strike. A right handed attacker will be close to the stone, and have limited options for striking. However, a left handed attacker will be able to exploit some of those architectural designs during the assault. A lefty climbing up the stairs will be able to strike with their less restrained arm.
This can also be seen with some external stairways, where the defender’s right arm will be out over open space, and free to move, while a right handed assaulter will have their weapon arm pressed up against the stonework.
While it’s not relevant to specifically melee weapons, hand dominance can be a major factor with modern firearms, to the point that some guns simply cannot be used with the wrong hand. The big offender are the controls. Safeties, slide releases, and mag ejects, can be ambidextrous, but it’s common to see those designed for a right handed user. Most firearms will eject their shell casings out the right side of the gun. This can be especially awkward if you are left handed. Some rifles (especially bullpups) cannot be operated off-hand, as they will gleefully pelt the user with spent shellcasings. Some grip contours will be uncomfortable, or unusable, if you attempt to hold it with the wrong hand.
This leaves the user with some very specific options. Some firearms can be reconfigured for left-handed use. (This can sometimes be achieved through configuring the weapon itself, though in other cases, you’ll need to replace specific components.) You can simply cope, and adapt your grip. Or, you can learn to operate the weapon off-hand. In my case, my experience with operating rifles right handed is simply because modifying the offending rifles wasn’t an option.
Many left-handed shooters will learn to operate firearms right handed, simply out of necessity. But it’s always nice when you’ve got the option to use a gun with your preferred hand, but, for a lefty, it’s not always an option.
So, does your dominant hand have any importance? Yes, some, but in the vast majority of situations it’s not going to matter that much. Being left handed isn’t that exceptional or unusual. It can affect combat, but it’s not a major consideration.