PEO application - how early shall I go?

[tj2014uk]

My initial T1G was granted on 6th April 2010 and the 2nd extension will expire on 19th April 2015.
So I become eligible for ILR on 9th March 2015 and wish to get PEO appointment by 20th March.

1. God forbid in case of rejection, will I still have appeal rights? As I'll still have approx.
20 days visa, is there a possibility that HO doesn't give me appeal right and tell me to re-apply?

2. Keeping in mind that 6th April is the last date of T1G extensions, it is very likely that March will be very busy for PEOs. Do you experts reckon HO will still be able to provide same day service for 'straight forward' applications?
I don't want to pay the extra fee for PEO if they transfer the application to the postal route.

3. How early Do I need to book an appointment for March?

Your thoughts will be greatly appreciated.