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. Similarly, if you want to bite with Str, you must use the Bite feature's damage die, since they are separate abilities that share the same moniker. So if your Monk's Damage die is a D8, but you want to use Str, it wouldn't be possible since you are invoking a feature that doesn't exist for the character.
- Either or; you choose which one. If it's Martial Arts, it must be Monk Dice + Dex. 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.