Some insurers will offer you a cash-in-lieu of repairs. Please enquire as to whether the car is being entered on MIAFTR, and if so, what the category is.
The majority of cars tend to be written off on purely economic grounds, but insurers are required by law to enter any vehicle declared a total loss on various databases.
As previously mentioned by NORMANTHEDOG, you may need a VIC check, although this isn't a requirement by law (before anyone argues with me, I investigate fruadulent insurance claims for a large insurer in the UK, specialising in total loss and theft claims, and I have a close working relationship with VOSA and the DVLA).
You may continue to drive the vehicle without having a VIC done, but the DVLA will not send you any tax reminders, and you will be unable to obtain a new V5 for the vehicle until the VIC is undertaken. This means that if you loko to sell it to someone, they will not be able to get a V5 in thier name, and this will come back on you.
Yes, you can get the car insured elsewhere, and no you don't need to inform them of this. If you get the car repaired yourself, please, please, please keep the repair documentation. Trust me that if you have another accident in close proximity to this one, and you choose to claim, this may be picked up on, and if you can't prove the car was repaired, it'll be a nightmare for you - this is what I do all day.
I would recomend putting the car through an MOT either before or after the repairs have been completed for piece of mind. VIC literally checks that the identity of the vehicle hasn't been changed, and not whether the vehicle is roadworthy or not.