To answer your first question, yes, rules for using a quarterstaff that way is already in the rules. They're under "double weapons" and "Two-handed weapons" in the PRD.
Double Weapons: Dire flails, dwarven urgroshes, gnome hooked hammers, orc double axes, quarterstaves, and two-bladed swords are double weapons. A character can fight with both ends of a double weapon as if fighting with two weapons, but he incurs all the normal attack penalties associated with two-weapon combat, just as though the character were wielding a one-handed weapon and a light weapon.
The character can also choose to use a double weapon two-handed, attacking with only one end of it. A creature wielding a double weapon in one hand can't use it as a double weapon—only one end of the weapon can be used in any given round.
and
Two-Handed: Two hands are required to use a two-handed melee weapon effectively. Apply 1-1/2 times the character's Strength bonus to damage rolls for melee attacks with such a weapon.
So yes, the core rules already provide for this.
(In fact - and I regret I don't have my rulebook here to check - I'm pretty sure that a wizard using a quartstaff is actually given as an example of this in the Player's Handbook.)
On your point 2, note that the greatsword, greatclub and other two-handed weapons you mention are weapons that require two hands to wield. They require two hands because they are larger and heavier than weapons that do less damage; Their being larger and heavier is also why they do more damage, not he fact that they require multiple hands to wield. Wielding a weapon two-handed does multiply the strength bonus you apply to damage, as mentioned above, but has no effect on a weapon's damage dice.
If you're looking to boost damage, the best way to do it is by boosting damage. To address your specific choices:
- Alert will slightly boost your damage by letting you deal damage when you would normally be surprised.
- Charger is mediocre at best before level 5, (which is one level away!), and just plain sucks afterwards. You have better things to do with your action and with your bonus action.
- Heavy Armor Master will boost your damage significantly, since it will increase your Strength score to 16, giving you +1 to attack rolls and damage rolls. It will also help you survive, which, in the long run, will increase your damage a bit.
As you've probably noticed, there is not a lot of feat support for sword-and-board warriors. Most of the damage boosting options go to 2-handed builds. One option you didn't mention is Shield Master. It allows you to knock foes prone with your bonus action, which can help your attacks hit, thereby boosting your overall damage. But, to be perfectly honest, you're far better off getting your Strength score up to 16.
Given that getting your Strength score up to 16 is the best boost to your damage that you can give yourself right now, it's definitely a priority. So I would recommend either Heavy Armor Master, or just taking a stat increase and getting your Charisma to 16 as well as your Strength.
"Wait, what? How does boosting Charisma boost my damage?"
Well, it doesn't directly boost your damage. But neither does Heavy Armor Master. And while in the short-term, Heavy Armor Master will make you a lot tougher, as the game goes on 3 damage will become less and less significant, and more and more monsters will bypass Heavy Armor Master anyway.
On the other hand, Charisma helps with almost everything else you do - your spells, your Channel Divinity, and, from level 6 onward, all of your saving throws and all of the saving throws of any ally standing near you. Where Heavy Armor Master will get less and less important, this will get more and more important.
So, final recommendation: Either take Heavy Armor Master, or take the ability score increase and get both Strength and Charisma up to 16. They're both good options, and they'll both boost your damage roughly equally. The main difference is that Heavy Armor Master is much better in the short-term, and Charisma is much better in the long-term.
Best Answer
No need to ban the bastard sword, because it's not part of D&D 5e yet. D&D Wiki is mostly full of things made up by random people on the Internet — the Bastard Sword you link to is some random person's homebrew. You would have to choose to add it to your game, not ban it.
Is it balanced? Meh, doesn't matter. You don't want it, so there's no reason to think about it further.
Avoid D&D Wiki. Use official rules references instead.
Unless you're specifically looking for homebrewed content ideas, you should probably avoid D&D Wiki like the plague, since it has so many problems. Actually, even for homebrew, it's notorious for being full of things with massive balance problems that aren't worth your time.
You can get the official Basic version of the rules from Wizard of the Coast's website at that link. You can also get the content online at one of the more respectable 3rd-party sites that republish the 5e System Reference Document (“SRD”), such as 5eSRD.com. WotC also runs D&D Beyond as their own 1st-party digital online version of the rules.
For rules and game material, the Player's Handbook, SRD, Basic rules, and/or D&D Beyond should be your first source, not some wiki or a Google search. You'll save yourself and your players a lot of headaches.