Here the answers to your queries:
1. No need to bring anything except your own clothes and some toiletries to meet your immediate needs. Almost everything available in your country is available here. LuLu is a major supermarket where you can find all sorts of Indian items. However, you will not be able to find "paan" or "daaroo" or other alcoholic drinks anywhere.
2. I guess it would be cheaper here.
3. There are just two companies in this field: Ooredo and Vodafone. Ooredo is said to have the upper hand. My personal choice is Ooredo.
4. Almost all banks provide equally good service. But among the leading ones are Doha Bank and Commercial Bank, While all banks have corresponding banks in most countries of the world, you will not be needing their service in transferring money to your country. There are other cheaper, faster and means which are legal. You will get to understand things better when you land here.
5. Qatar has three types of public transport service. The country has just started a railway service which is very economical. However, almost all major parts of the city is linked through a bus service known as Karwa. The buses operate on fix timings usually at intervals of 30 minutes. The services are good, reliable and also very economical. It costs QR 3 for a single trip within the city from any one point to the destination. There are also taxis readily available.
Always keep your Qatar ID card with you when you leave home as police may stop you to check. There is no such thing as bribing a policeman or any government official. Don't ever attempt it. Don't land up in a fist fight with anyone. Restrict your anger to just verbal exchange of (hot) words in such cases. Don't bring any medicines unless they are supported by a prescription. Don't get yourself involved in "love affairs". Be a law abiding citizen. Almost all supermarkets have CCTV cameras installed and you will be watched.
Trust this helps.