The place where the railroad crosses the 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 ;-)

In the UK, this is referred to as a level crossing, the name implying that the road and rails are on the same level, as opposed to one going over the other via a bridge or tunnel.

The red and white things are barriers. They lower or come down to block the road.

So your example sentence becomes:

"At the level crossing, when the barriers finally came down, it was obvious I would be late for the party."

British level crossings which have automated barriers also have red and yellow lights near the crossing. The red lights start flashing a few seconds before the barriers come down, and stop flashing after the barriers have gone up.

In the US, these kinds of railroad crossing are often referred to as "grade-level crossings" or just "grade crossings" - "grade" means that the railroad and car road are at the same height (no overpass or underpass).

People in Western Canada refer to them as "Railroad Crossings", so:

"At the railroad crossing, when the gate finally came down, it was obvious I would be late for the party."