could this work?
No. All weapons with the "heavy" property have the two-handed property as well. Here's exactly what each property does (PHB 147):
Heavy. Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon's size and bulk make it too large for a Small creature to use effectively.
Two-Handed. This weapon requires two hands when you attack with it.
The heavy property itself isn't a problem here, but the two-handed is. Since all heavy weapons are also two-handed, you must use two hands to wield them. The enlarge spell doesn't give you the ability to wield a two-handed weapon in one hand, even if that weapon wasn't enlarged with you. Your DM might allow it, but it's certainly not part of the Rules As Written.
So what can you dual wield, then? Here's the rules on two-weapon fighting (PHB 195):
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in one hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Based on this, you can only dual wield light melee weapons. You don't even need the two-weapon fighting style, since all the fighting style does is allow you to add your ability modifier to the bonus action attack. There is a way to get around the "light" limitation, namely the Dual Wielder feat (PHB 165, emphasis mine):
You master fighting with two weapons, gaining the following benefits:
- You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand
- You can use two-weapon fighting even when the one-handed melee weapons aren't light
- You can draw or stow two one-handed weapons when you would normally only be able to draw or stow only one.
This feat allows you (among other things) to use any one-handed melee weapons for two-weapon fighting.
In conclusion, you're not going to be two-weapon fighting using greatswords, even if you're enlarged. The closest you're going to get would be two longswords with the Dual Wielder feat
Side note: If you want to do two-weapon fighting purely from an optimisation perspective, you may want to reconsider. It is generally accepted as one of the weaker playstyles, as it uses your bonus action to be effective (needed to cast smite spells on a paladin, for example), and still can't quite compete with great weapon fighting, especially if you factor in the Great Weapon Master feat. It doesn't fall completely behind other playstyles, so if you want to do it because of a character concept, go ahead. But if it's exclusively for optimisation, it might be better to just use a single greatsword.
If you can fit it through the opening, you can put it in there.
This would effectively make your Bag of Holding a wonderful trap/container for your Genasi.
As for opening it from the inside, rules do not state you can or can not. Lots of people have asked this before, but I personally do not feel that you can open the bag from an extra planar space because you aren't technically inside a bag, you're actually on a different plane. To me, that means you can't manipulate the opening of the bag.
But that's my opinion on it.
Material, somatic, and verbal components are only required during the casting of spells, not during the duration. Your character can't shoot a bow while casting a spell with material or somatic components, but once the casting is done, your character's hands are free to do whatever you want while the spell is ongoing.