Is DbUpdateException thrown by the Execute extension?

1. no it doesn't; if you want that, you should be doing things like select @@rowcount after the delete in the tsql, using ExecuteScalar<int>, and checking that the number returned is 1. If you want timestamp checking, you would include that in the where clause and as a parameter

2. never, never, never concatenate inputs into the data part of SQL; it creates a SQL injection risk, and destroys all query / operation caches - the first reason is all you should ever need, though. There is a bit of a caveat here around things like table names, but you should be white-listing there. Note that dapper supports full parameterization (easy parameter handling is one of the main reasons to use it!)

For example, I would be doing something like:

public YourType(string tableName) { // constructor
    WhiteList.AssertValid(tableName); // throws if not allowed
    deleteCommand = $"DELETE FROM [{tableName}] WHERE id = @id; SELECT @@ROWCOUNT;";
}
private readonly string deleteCommand;
public virtual void Delete(int id)
{
    var connection = dataContext.GetDatabase().Connection;
    int count = connection.ExecuteScalar<int>(deleteCommand, new { id });
    if(count != 0) throw new DbUpdateException();
}

Alternatively, use an add-on tool like dapper-contrib that does all this stuff for you.