The name for a barrier where a railway line crosses a road

I can't think of many terms for this... all I can think of is a crossing or railroad crossing (Merriam Webster).

As for the "red-white" thing I would call it a gate or railway/railroad crossing gate - very descriptive terms, but that is what I would use.

As for your sentence I would say "At the railroad crossing, when the gates finally lowered, it was obvious I would be late for the party".

Check out the "Rail crossing warning signs" from Wisconsin, there might be other resources from other parts of the English speaking world?

AFAIK, barrier is more common in British English for the "gates" talked about above. Australian English i have heard of booms - but about these two I'm not the one who should be asked ;-)

The Wikipedia article on this subject begins,

The fez (Turkish: fes, plural fezzes or fezes[1]), or tarboosh (Arabic: طربوش‎ / Egyptian Arabic pronunciation: [tˤɑɾˤˈbuːʃ], ALA-LC: ṭarbūsh) is a felt hat either in the shape of a red truncated cone or in the shape of a short cylinder made of kilim fabric.

A Google image search also confirms that these two words are used interchangeably for the same type of hat.

It should be noted that variations in the style and structure of traditional hats like the Fez are common and trends will often form geographically. For example in Morocco there is a common type of shoe that when purchased in the south of Morocco will have a different style than if purchased in the north.

Because the words will also divide themselves on geographic boundaries, based on both social pressures and the etymological differences between the words, it would not be surprising to find people who will use the words to refer to different styles of hats.