what happens to your disability allowance and state or private pension from the UK, if you decide to live 6 months in Philippines and 6 months in the UK?

State pension is not affected, and increases are payable, even if you remain here. Private pension is not affected. Best to have both paid, into your UK bank account. The only problems I envisage, is when you need a Certificate Of Continued Existence from either. As it might go to the address, that you are not living in at the time.
Disability allowance, I have no idea at all.

Yes the paperwork for pension continuance is inconvenient. If they stop the payments then a phone call will see them restarted.

Not sure if they will e-mail the forms as a standing instruction for your to print off, complete and return as a hard copy.

what do you suggest I do regarding this certificate?

are proof of life forms available for download??
how does one get them? can it be requested by calling IPC?

Suggest you call IPC and ask