Log all errors in queries

If you are just looking to log this information, you can set up an Extended Events session and capture the error_reported event. Here is an example:

CREATE EVENT SESSION [ErrorCapture] 
ON SERVER 
ADD EVENT sqlserver.error_reported
(
    ACTION
    (
        sqlserver.client_hostname,
        sqlserver.database_id,
        sqlserver.sql_text,
        sqlserver.username
    )
    WHERE 
    (
        [severity] >= (11)
    )
) 
ADD TARGET package0.asynchronous_file_target
(
    SET filename=N'C:\SqlServer\Testing\ErrorCapture.xel'
)
WITH 
(
    MAX_MEMORY=4096 KB,
    EVENT_RETENTION_MODE=ALLOW_SINGLE_EVENT_LOSS,
    MAX_DISPATCH_LATENCY=30 SECONDS,
    MAX_EVENT_SIZE=0 KB,
    MEMORY_PARTITION_MODE=NONE,
    TRACK_CAUSALITY=OFF,
    STARTUP_STATE=ON
);
GO

ALTER EVENT SESSION [ErrorCapture]
ON SERVER
STATE = START;
GO

To test this out, here is a test error with RAISERROR():

raiserror('This is a test error', 2, 1);
go

Then by looking at the XE log through the sys.fn_xe_file_target_read_file system function, you will be able to see all of the logged errors.

Yes. Monitor for Event Notifications in the Errors and Warnings Event Category. Specifically, the Exception event will be raised for each error. For example, based on the code in MS SQL Server third party transaction blocking monitor tools:

use msdb;
go

create queue [errors];
go

create service errors on queue [errors] (
    [http://schemas.microsoft.com/SQL/Notifications/PostEventNotification]);
go

create event notification [errors]
    on server for EXCEPTION
    to service N'errors', N'current database';
go

If we check the queue, we'll see all exception reported:

receive cast(message_body as xml) from errors;


<EVENT_INSTANCE>
  <EventType>EXCEPTION</EventType>
  <PostTime>2013-02-19T13:52:21.443</PostTime>
  <TextData>Violation of PRIMARY KEY constraint 'PK__x__3213E83F31271CCE'. Cannot insert duplicate key in object 'dbo.x'. The duplicate key value is (1).</TextData>
  ...
  <HostName>...</HostName>
  <ApplicationName>Microsoft SQL Server Management Studio - Query</ApplicationName>
  <LoginName>...</LoginName>
  ...
</EVENT_INSTANCE>