We can't really answer all the questions you are asking as we don't have any facts - we can only guess like you. You, or someone who provides care for you, will need to investigate this with both the DWP and the Medical Services company that carries out the medical assessments. These are 2 separate organisations. You need to contact the medical services company (rather than the DWP) and ask why your GP's request for a home assessment was ignored (as per Monic's advice). If your GP's letter was submitted by you along with the completed ESA 50 form then it will all be with the medical services company as the form goes direct to them.
You need to explain that the letter inviting you to attend a medical was issued by them before you had even returned your completed ESA-50. Ask them why this has happened. They may or may not have an explanation. It may be a simple matter of a computerised list of letters being automatically generated and sent out on the assumption your form was in on time.
Regardless of any explanation, you are going to have to have a medical so you need to move on and explain that your GP has specifically requested that you have a medical done at home - tell them this GP letter was with your completed ESA-50 so they must have it. Ask them if they will now agree to provide that medical at your home and if not, ask why not. In view of the very tight timescales, you may have to be prepared to go to the medical at the medical services company's office - I don't know why you and your GP think a home assessment is a better option for you but is it possible for you to go to the medical services company's office for the assessment if you need to?
The risk is that with very tight timescales, all of this mis-communication and all of the confusion over your request for a home assessment, if you do not turn up on the allotted date at the office, they will cancel your claim. If you are absolutely certain that whoever is speaking to the medical services company and the DWP on your behalf is 100% sure that they are sorting things out with the right people and that your claim won't be cancelled if you don't turn up on the allotted date then it might be worth the risk of holding out for a home medical. If you absolutely cannot under any circumstances go to the office for the medical then that should be clear from your GP's letter and it should be an easy matter for your caregivers to sort this with the medical services company.