No to all three.
While the unarmed strike is in the weapons table on (some editions of) PHB 149, the errata states that it should not count as a weapon. Therefore, attacking with an unarmed strike does not count as an attack with a one-handed weapon.
Moreover, while the grapple and shove special attacks are Attack actions, the rules regarding them make no reference to using a weapon. Because they don't use a weapon, they don't qualify for the "...with a one-handed weapon" clause.
Your Attack Action and Bonus Action attack are not interchangeable.
In order to use a grapple or shove, you have to use your action to take the Attack action, which is different from simply making an attack (PHB 195, emphasis added).
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple.
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.
While you can use one of your attacks from multiattack to do these things, so that you don't have to use your entire Attack action, these special attacks only work in the context of an Attack action.
The feat allows you to use your bonus action to make an attack, but it is not an "Attack action". For an example of the distinction between an "attack" and an "Attack action", you can look to the passage on Two-Weapon Fighting on PHB 195:
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.
See how there's a distinction between the Attack action and the bonus action attack?