I'm not happy about how my insurance company settling my claim without my permission, can they do this? 

Yes, the have the right to SUBROGATE the claim. This means they can take over recovery of the claim and look to minimise any future costs. This is usually done when liability cannot be definitively proven. 

BUT, if you feel like you have sufficient evidence then you can absolutely questions this. Contact your insurance company and discuss this with them. If it becomes necessary and you are still not convinced then log a complaint. You also have the right to access all the information on your claim- redacted for GDPR. 

My insurance company wont give me a bigger courtesy/hire car. Why can I get something something like for like?

What your insurance company offers you is most likely what you have as per your policy coverage. If you selected that coverage for its lowest price then you may well be offered a vehicle lower down the spectrum of available vehicles.

Note: there is a difference between a courtesy car and a hire vehicle. A courtesy vehicle comes from your garage in order to keep you on the road during repairs- usually a 1 litre, 3 door vehicle. Where as a hire vehicle is supplied by a hire company is generally newer but, unless agreed on your policy, is not usually like for like.  

But I was given a better vehicle on my last claim, why not this time?

There is a good chance that the liability for you previous claim was a confirmed non-fault and they where able to offer you a form of credit hire which is paid for the third party insurer directly. This can usually get you a vehicle closer to your own. 

My insurance company is not doing a thing to progress my claim. I am constantly having to follow up with them. What can i do?

You have two options:

1- log a complaint pointing out exactly what they have siad they will do vs what they have done and how much inconvenience/stress this has caused. When you do this the are legally bound to log, investigate and contact you with the results of their investigation. All of this is logged for review/audit. When they come to an acceptable resolution, make sure you have them follow up on the complaint every two weeks until you are satisfied. 

2- If they are ignoring your issues then contact the financial ombudsman who will take a full and independent look at your claim. 

Do i have to pay my excess, even though I'm not at fault?

In a word, Yes. It is party of your insurance conditions that this is paid upon the completion of repairs. 

UNLESS liability can be conclusively proven and/or accepted by the third party, and you have all details of the other vehicle. 

But the third party insurer has offered me services to repair my vehicle.

If you prove a letter/email stating that they have accepted lability then it is down to your insurer to decide.

My insurer has offered me a low amount for my vehicle, is there a way to get the amount increased?

Potentially, yes. It all comes down to the justification for your own valuation.

The similar vehicle prices needs to be exact make and model of you vehicle, the same registration bracket (e.g. 16 plate) and with approx. 10% of the milage. AND from a reputable/independent source. 

Additionally, if you have any recent major work done to the vehicle may increase he value if it can be proven it wasn't done too many miles ago.