For a conservatory to be classed as such, you need a minimum of 50% glazed walls. So if by adding the solid side walls your project, the glazed is less than this figure, then it will no longer be classed as a conservatory and you need to speak to your local Building Control Officer. If it's still higher than 50%, then it's still a consevatory and is exempt from the regs.
None of this has anything to do with planning permission. If it's permitted development, then that's it, no matter what you class it as.
All councils work on the same rules, but sometimes you need to clarify certain interpretations.

hi,, you proberbly dont require permission but will need to involve b/c to see its done correctly and is save with whats below ground........

hi, always a safe bet to involve building control,
would imagine footings would need to be deeper for high walls but only a trial hole will conclude this.

Hi, all councils have different opinions on what is acceptable. As previously stated you would be best to involve them as they would still want any alterations to meet building regulations.

The works you describe will need planing permission/permitted development, a call to your Local Authority will confirm this.