Concerning the thread you posted... Aliases is not a "function" of TestComplete versus TestExecute... it has to do with the project you are running. Based upon your screenshot, you are mapping your objects using namemapping which means your project has Aliases... but you haven't responded/resolved whether or not the project has been properly transferred over to the machine along with the appropriate namemapping file. If not, then yes, you will get the "Aliases" not supported error.
That said... if you're getting an "object does not exist" error, that usually points to an issue in timing where your scripts are attempting to interact with an object that has not been instantiated. Keep in mind that automated tests playback a LOT faster than human interaction... that, and human interaction has this automatic reflex to not try and do something with an object that has not yet been presented.
Read up on using WaitNNN methods like WaitChild or WaitAliasChild to actually check to see if your object exists before you attempt to click on it or interact with it.
