Currently, I am changing my job, and get some questions about renew my Tier 2 (general) working visa, please help

1. When should I apply?
My situation is: I am currently working for company A and my contract will end on 18th May 2015, and my new job will be started on 19th May 2015 in a different company B as we agreed. My current Tier 2 (general) visa will end on 18th May 2015.
I want to apply my Tier 2 (general) visa extension in person using the premium service, but not sure when I should do this.
I have got the COS from the new company B today. The COS is valid from 13th Mar 2015 to 13th June 2015. In the COS, it clearly states that my new job will start from 19th May. The nearest date I can book for the premium service is 6th Apr 2015. If I go the the visa center on 6th April, presumably the case work will give me the decision on the same day. If it is successful, I assume the case work will give me a new Tier 2 visa valid from 6th April or 7th April.
My old biometric card will be taken away on 6th May and my old Tier 2 visa has gone. Can I still working for my old company A? Because my contract will end 18th May, if my new visa is valid from 6th April, does it mean that I can't work for company A anymore?
In this case, should I apply in person in May some time, for example, some time near 18th May?
Or I can go for 6th April and ask the case work to assign me a new visa starting from 19th May? Sorry, I am confused about what I should do.
2. My current job in company A is not in the shortage list; however, my new job in company B is in the shortage list. For this case, should I apply extension of normal Tier 2 (general) or Tier 2 General (up to 3 years) - shortage occupation?
The price for this two are different, normal Tier 2 (general) will cost £1,001 (application in person), and Tier 2 General (up to 3 years) - shortage occupation will cost only £828 in person.
3. Recently, UKBA renamed into UKVI, should I use the old website system to do the online application form and premium service booking or the new one? The new one is called beta trial version.
Old one: https://apply.ukba.homeoffice.gov.uk/iapply.portal
New one: https://visasandimmigration.ipt.homeoffice.gov.uk
Actually, I couldn't find the premium service option in the new system.
4. For a person like me applying extension of Tier 2 (general) in the UK, should I provide the results of 'Tuberculosis Test'? I have already stayed in the UK for about 3 years. I have been to Thailand and India for a holiday for two weeks last May.
Thanks very much if you can share some of your experiences and give me some guidance. Have a nice weekend.
