One website targeted to multiple country and cities is it possible?

Hi,

Greetings, I Would like to get few great guidance and help on website division for seperate country,

So to start with, I am launching a membership website, and we are planing to launch this website for multiple country and cities, so i am bit confused about few things,

We are launcing in,

1. US, ( Chicago and Phoenix )
2. Singapore
3. Australia
4. India ( Bangalore, Delhi )
5. UK, ( London, Nottingham )

So these are the places we want our website to appear on the local search results, map results also on organic listing for the city based search like "Product abz in London" or "Product abz in Chicago"

so this is how we want our website to get ranked, but I am bit confused on few things,

Please check this image here --> http://goo.gl/a77vt0

At the moment how I am planing to sort this out is based on having my website TDL which is

SEO optimization for US

www.MySite.com ---> will be for US
www.MySite.com/Chicago ----> For Chicago, US
www.MySite.com/Phoenix ---> For Phoenix, US

SEO Optimization for Singapore

www.MySite.com/Singapore ---> For Singapore

SEO Optimization for India

www.MySite.com/Bangalore ----> For Chicago, US
www.MySite.com/Delihi ---> For Phoenix, US

So I want to know if that is fine to do SEO for those separate innerpages, with in the same main website? or we need to have seperate domain name for each different country?

like example

www.MySite.co.uk

www.MySite.sg

www.MySite.in

www.MySite.co.au

So that is how i need to optimize it for different countries? since my main goal is that I want to rank those links to those country so I am not concerned that it should look professional but as long as it would be better to do optimization I would follow the guidelines,

So would you please guide me with this help please?

Since I want to rank our website in multiple country but which was would be better to do it?

Thanks
Nick