Can the PC continue with further secret ballots? If there isn't any provision in standing orders for a secret ballot, then no. Double check your standing orders though - in model standing orders this process is covered in 8. In any event, any councillor may request a recorded vote. If you do not want there to be a secret ballot, request a recorded vote.
Will they have to change standing orders? If standing orders do not allow for a secret ballot then yes, they would need to be changed to permit a secret ballot. (There is a rare exception to this which would be if standing orders allowed a motion without notice to hold a secret ballot). Again, any attempt to hold a secret ballot can be defeated simply by any councillor requesting a recorded vote.
Can I have an original vote and a casting vote in the event of a tie? If you are in the chair and a member of the council then yes, you have an original vote and a casting vote - just as you would for any other motion. The chair of a meeting always has a casting vote. However, if the chair of a meeting is no longer a member of the council (eg the outgoing chair after an election) they do not have an original vote.
Out of curiosity, it is an interesting time of year to be raising this question if you are the existing chair. I'm wondering (if you don't mind my asking) why this has come up now?