The Vulcain Cricket is another watch associated with presidents.

Definitely a Rolex Datejust. Looks like a 34mm or 36mm.

Yes it’s a real watch.

Can’t go wrong with any of those. If you want something a little different, check out Torsti Laine. The finishing on his customized Laine V38 with triple engine-turned (by hand) guilloche watches blows these 4 out of the water. Only con is that you’ll have to wait 8-10 months for delivery.

Maybe a pre-owned Tudor Black Bay (the ones with the plain/non-rotating bezels) or an Omega Aqua Terra. Some of the older ones can be found for right around 3k.

Fugazi.