The above question reasons that an unarmed strike is defined as:
"Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (PHB 195, errata v1.0)
The Lizardfolk's Bite attack is defined as:
Your fanged maw is a natural weapon, which you can use to make unarmed strikes. (VGtM 113)
Emphasis mine, notice the plural. Additionally,
When you use the Attack action with an unarmed strike or a monk weapon on your turn you can make one unarmed strike as a bonus action. (PHB 78)
Therefore, you can supplement it in lieu of any type of unarmed strike, not just your fists, kicks, headbutts, ect.
To further answer your question:
You can use the D6 damage die until it is weaker than your standard Monk damage die; you can even bite with Dex, but you must use your Monk's damage die, since at this point you're simply biting as flair. You can also use Str, as the Martial Arts feature says
You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons. (PHB 78).
Emphasis mine. It does not say that you must use Dexterity to gain the benefits of Martial Arts. As such, your bite effectively becomes a flavor feat at 5th level, since you can deal 1d6+Str (or +Dex) with any unarmed strike.
- Either or; you choose which one. If it's Martial Arts, it must be Monk Dice + Dex or Monk Dice + Str. If it's Bite, it must be a D6 + Str.
- It's a separate feature, but should work with any Bite you do, as long as you have a free Bonus Action.
- Yes, but only using Martial Arts, aka Monk Dice.