Is it necessary to use same companies for enzymes in the protocol, such as Fermentas for BamHI or Invitrogen for polymerase (Accuprime Supermix 1)?
NEB BamHI would work just as well. Otherwise, we would recommend some testing before changing companies.
Can the Invitrogen platinum Hot start PCR mix be used over the Invitrogen AccuPrime SuperMix I for the final PCR amplification stage?
Certainly. We ourselves have lately switched to Phusion HF Master mix, which is more efficient (it brings down PCR by several cycles, without much change in library quality).