What does SQL Server do with a timed out request?

The timeout is something that happens on the connection, not the running query.

This means that your BEGIN CATCH will not execute in the event of a timeout, as the query has no idea about it.

Write your cleanup in C#, in a catch(SqlException ex) block (testing for a timeout).

The timeout is enforced by ADO.NET. SQL Server does not know such a thing as a command timeout. The .NET client will send an "attention" TDS command. You can observe this behavior with SQL Profiler because it has an "attention" event.

When SQL Server receives the cancellation it will cancel the currently running query (just like SSMS does when you press the stop button). It will abort the batch (just like in SSMS). This means that no catch code can run. The connection will stay alive.

In my experience the transaction will be rolled back immediately. I don't think this is guaranteed though.

TL;DR: A timeout in ADO.NET behaves the same as if you had pressed stop in SSMS (or called SqlCommand.Cancel).

Here is reference for this: https://techcommunity.microsoft.com/t5/sql-server-support/how-it-works-attention-attention-or-should-i-say-cancel-the/ba-p/315511