The litigator is the court lawyer who will take this to court. Issuing proceedings is the process of going to court and serving papers on the other party. If this isn't done at this stage your case would become time barred and you wouldn't get anything.
No, they should do that and send it for your approval.
The solicitor would generally sign the form on your behalf.
Anything relevant to the claim such as medical reports in your case for example. Your litigator will deal with what is required at the relevant time.
If the claim form is wrong you have to tell the solicitor and ask for a revised form.