A claim could be issued by a litigation friend for a minor or by an attorney.  Otherwise, the court would expect the claim to be issued by the person with the claim.

In the first instance there is an obligation on the parties to a dispute to attempt resolution without litigation via Alternative Dispute Resolution (ADR).  If this is not possible court proceedings will be necessary.

If you are an interested party (executor or beneficiary) you will probably receive a letter about the dispute and be invited to take part in ADR and/or subsequent court proceedings.

Yes. A Will can be challenged on the ground that at the date instructions were given and/or the Will executed the testator lacked capacity.  Medical records, witness evidence and expert psychiatric evidence is likely to be needed to prove the claim.  

They will hear cases such unfair dismissal and wrongful dismissal, claims for redundancy payments and/or unlawful deductions from wages and discrimination claims under the Equality Act amongst others.