How does the Dual Wielder feat interact with versatile weapons?

I have a level 4 Paladin with the Dual Wielder feat (PHB p. 165):

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 you are wielding aren’t light.
  • You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

He does not have the Two-Weapon Fighting fighting style. He is wielding a longsword in one hand and a handaxe in his other (a thrown weapon).

Can the paladin throw the handaxe, switch grips on the longsword to hold it two-handed, and then use his bonus action from Two-Weapon Fighting to swing the longsword as a two-handed weapon?

If yes, can he then end his turn by drawing another handaxe (2nd free object interaction for Dual Wielder, and assuming he has another axe to draw)?