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To be specific, I'm planning on taking the XGtE Kensei monk to level 17, then dipping Battle Master fighter level 3 for the Dueling fighting style and maneuvers. At monk level 17, my Martial Arts die is a d10, and I can use my Martial Arts die with my kensei weapons.
Does this mean I can get my longsword to a d10 and get the bonus from Dueling if I wield it one-handed?
As you have pointed out, your martial arts die at monk level 17 is d10, and if you pick longsword as one of your kensei weapons, then you can deal a d10 with that weapon. From XGtE, p. 34:
Kensei Weapons. Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don't already have it. Weapons of the chosen types are monk weapons for you. Many of this tradition's features work only with your kensei weapons.
And from Martial Arts class feature, PHB, p. 78:
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
If you are wielding the longsword in one hand, regardless of how the damage die has been modified by the martial arts die, you can add the +2 to the damage as per the Dueling fighting style (PHB, p. 72):
Dueling
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Your longsword (wielded in one hand) will deal d10 + DEX + 2 damage.
However, also note that even without the Way of the Kensei archetype, you would be able to deal the same amount of damage if you were using a weapon that was already a monk weapon, such as a shortsword, dagger, quarterstaff, etc, so long as you wield it in one hand (for the +2 from Dueling).
Yes, you can apply your Martial Arts die with your Kensei weapons. Actually, you would be able to do so with any monk weapon.
The Path of the Kensei feature makes eligible weapons into Monk Weapons.
P. 34 XGtE
Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don’t already have it. Weapons of the chosen types are monk weapons for you. (...)
P. 78 PHB
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.
Dueling adds +2 damage to attacks with melee weapons when wielded 1-handed without a weapon in the off-hand.
P. 72 PHB
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Now, is this optimal, or even good in terms of damage? Well, no. That bonus damage only applies to 2 of your 3/4 attacks per round. Action Surge only doubles 2 of your 3/4 attacks. A fighter is throwing out 4 APR minimum at L20, with all the other goodies Fighter gets.
If you use a shield in your off-hand, you can't use your monk features, and a Longsword is no better than a dagger as a monk weapon (both will use a D10). Essentially, you are a monk with +2 damage (on 2 of your attacks), a few extra HP, Action Surge, Combat Superiority Dice, no Empty Body, and 1 less ASI. You miss out on Perfect Self, but it's kind of rubbish anyways.
