I cannot make sense of how this works.

1. The damage effect of the Unarmed Strike option includes language about making an attack roll, but the grapple and shove effects do not; but
2. damage, grapple, and shove are all listed as effects; and
3. Punch and Grab from the Grappler feat implies that an unarmed strike attack roll occurs before choosing the effect (and with the feat you can choose both damage and grapple rather than only one effect).

Do all unarmed strikes require an attack roll, after which grapple and shove also require a save? Or is an attack roll required only if choosing to damage? And if the latter, why does Punch and Grab imply that effects of Unarmed Strikes are chosen after the attack roll?

It looks to me like either it was a mistake to include "make an attack roll" under the damage effect OR the Punch and Grab description is poorly written.

If you make an unarmed strike, you have a choice of three options. One is an attack roll. The other two are saves.

Punch and Grab could be better written, but damage is the only option to make an attack roll so it's the only one that can hit.

So, Punch and Grab should be written something like this:

Punch and Grab. When you make an Unarmed Strike as part of the Attack action on your turn, you can use both the Damage and the Grapple option. 

Saying "make" rather than "hit a creature with" would better represent the rules as intended.

RAW Punch and Grab must be used against a creature specifically, such suggestion would not prevent an object, or a location.

Quote from Szabot >>

So, Punch and Grab should be written something like this:

Punch and Grab. When you make an Unarmed Strike as part of the Attack action on your turn, you can use both the Damage and the Grapple option. 

Saying "make" rather than "hit a creature with" would better represent the rules as intended.

I disagree with this interpretation.  The Punch and Grab option of the Grappler feat says "When you hit a creature with an Unarmed Strike as part of the Attack action on your turn..."

A hit requires a successful Attack Roll.  If you hit, then you roll Damage, and you have the option to force the target to make an STR or DEX save to avoid being Grappled. Neither the Grapple nor the Shove options use an Attack roll, so they cannot deliver the hit required for Punch And Grab. "make an Unarmed Attack" would be an incorrect interpretation, and would allow the Grapple attempt on a miss.

No attack rolls:

https://www.dndbeyond.com/sources/dnd/sae/sage-advice-compendium#SAC-Combat20

1. Does a grapple or shove trigger a Battle Master Fighter’s Riposte? No. Many game effects specify triggering when you are hit or missed by an attack roll; the Grapple and Shove options of an Unarmed Strike don’t use attack rolls.