Whose responsibility is it to discharge/approve requirements and when should the wording be agreed?

There are no express provisions in the Planning Act 2008 (PA 2008) or related secondary legislation dealing with which body should approve/discharge requirements.
The absence of such prescribed provisions under the PA2008 regime means that applicants have the opportunity to draft and statutory consultees suggest requirements on the basis of who they consider is the most appropriate body to discharge individual requirements and make a case to the Examining authority (and the decision maker) as to why the DCO should be made in this form.
As you know, the suggested requirements in the Infrastructure Planning (Model Provisions) Order 2009 (MPO) are drafted on the basis of the IPC discharging these, usually following consultation with another body such as a LPA. Since the MPO is not mandatory (see s38(3) of PA2008), applicants could though draft their DCOs on the basis that requirements are to be discharged by other bodies such as a LPA rather than the IPC.
Applicants should though have regard to relevant guidance published by DCLG and the IPC. IPC guidance note 2 (paragraph 20) states that the model provisions "are intended as a guide for applicants in drafting orders, rather than a rigid structure." Applicants should though explain in the explanatory memorandum (ExM) "if and why any provisions in the draft order diverge from the model provisions" (DCLG application form guidance, paragraph 20).
Paragraph 70 of the DCLG Guidance for Local Authorities advises that local authorities should recommend to the applicant and the IPC appropriate requirements "including any subsequent approvals to be delegated to local authorities for decision". The implication of this being that the discharge of requirements (other than those relating to the marine environment) will generally be dealt with by local authorities rather than the IPC.
Paragraph 71 of the DCLG guidance for local authorities refers to the Government's intention to amend the MPO with regards to discharging requirements. However, the Government has since announced that it will be revoking rather than amending the MPO. The intention is that DCLG Guidance dealing with this matter will probably be published at the same time as revocation of the Model Provisions.
s.120(5) of the PA2008 allows a DCO to apply, modify or exclude a statutory provision. This could, for example, include the appeal procedure provisions in the Town and Country Planning Act 1990. It is for applicants in the first instance to draft requirements, and other provisions, in their draft DCOs. The IPC is therefore unable to provide you with any model provisions beyond those contained in the MPO. You may though wish to look at draft DCOs in respect of accepted applications for development consent on the IPC website.
Notwithstanding which body(s) is to discharge requirements, our advice is that applicants should be engaging with relevant local authorities, highway authorities and other statutory consultees at the pre-application stage, as part of their s.42 consultation, in order, amongst other matters, to identify what requirements those consultees consider may be necessary and if possible agree the wording of these before an application for development consent is submitted.