Errors
The MyOwnFreeHost API doesn't use any error codes, and will always return a 200 HTTP code. This is less than ideal, but it is what it is.
Ticket Creation
Check the returned string to see if it contains SUCCESS :. If it does, the ticket creation was successful. If it does not, the request failed and the string contains the error message.
Ticket Reply
Check to see if the returned string is exactly SUCCESS (be sure to use PHP's trim(), Python's .strip(), etc. first). If the string is not exactly "SUCCESS", the request failed and the string contains the error.
"Information" Endpoints
No good way to detect errors, you probably don't ever need to run these on production/live environment anyways.
"Accounts" Endpoints
Take a look inside "result" -> "status" (JSON API) or the <status> tag (XML API). Run trim()/.strip()/etc., then check if the result is 1 (don't use strict comparison, or cast to int first). If status is not 1, the error can (usually) be found in "result" -> "statusmsg" (JSON API) or <statusmsg> (XML API).
"Domains" Endpoints
These are all different. See the endpoint's page for additional information on what each one returns.