start date of CoS is early than brp

Hi,
I am expected to start to work in the University of Gloucestershire from 19 October. And this university have already assigned me a certificate of sponsorship in which the valid time is from 19/10/2020 to 18/10/2023. Also I have just submitted the new Tier 2 visa application, but this visa decision by UKVI will need perhaps 2 months. So I will not get my new visa before 19/10/2020.
I am currently employed by the University of Manchester, and my current BRP (visa) expiry date is 6 January 2021.
Now, I have some questions:
1 Before I get the new visa, I should remain to be employed by the University of Manchester even after 19/10/2020 which is the starting date of the new CoS, is it correct?
2 Before I get the new visa, I can't start to work for the University of Gloucestershire, is it correct?
3 Before I get the new visa, if the University of Manchester cancels my current visa before 19/10/2020, I must not stay in the UK and must leave UK, is it correct?

1. Your choice if you leave or not.

2. Correct

3. No. They only I form ukvi. Only UKVI has the authority to curtail your visa. This process usually takes some time and you will be 60 days to apply for a visa or leave the UK.