Doors – Does such a door knob/handle exist that has the latch bolt installed on the outside of the door

You're looking for a Door Shoe

A Door Sweep

Or a Threshold with a vinyl seal

Or a combination of these.

These products are typically used to weather seal an exterior door from the elements, but should work just as well in your situation. They should all be available at your local home improvement store.

Identification in order, P, Half-Round and Quarter-Round

Referred to as door seal or weatherstrip.

Commonly available for aircraft door seal and older automotive trunk and door weatherstrip.

You can get EPDM variants for home door seal use from WeatherKing or MD.

The larger stuff used in automotive can be obtained by the roll from industrial supplies like Grainger.

The US has largely abandoned this stuff as it's easy to snag, miserable to mount and unless made from EPDM rubber, not long lived in sunlight.

The vinyl clad "V" seals that replace it for door casement sealing need only a narrow saw kerf (hence their other name, "Kerf Seal") to be pushed into for mounting, no weatherstrip cement to get all over the place or peal off tacky tape that quits sticking to the surface so it falls off.

With proper construction so it has full contact and proper compression, it is both air-tight and sound-tight. When worn or damaged, replacement is easy, just gently pull from the slot and press the new strip back in.

You also might search for kerf seal weatherstripping (Amesbury is one such manufacturer) as there are a lot of other cross-sections available that might be more what you're looking for. Having to just cut a slot and have it both readily available and easily replaceable via saw slot is something that will be desirable down the road.