kiranputhran 1> Can we keep an application coded in j2me constatly running in a phone memory which can constatly connect to web server? Can we do it using Socket or gprs or any other.
If you mean running in the background, then no. However, you can try reading Push register. It can call your app when SMS arrives. Maybe there is some further functionality, check it
2> If we can keep an application working on background can and that application collect the data from its phone book and automatically send and sms or data to the server?
MIDlet can't access phone book due to sandbox limitations. Only place where it can "half-interact" is when you add TextField of type PHONENUMBER. Some phones will allow you choose number from phone book.
3> Can MIDlet allow to access and modify "Phone Book"?
No. MIDlets are basically yhings on their own. Phone is just a container
if no any other means?
Use Symbian programming. Your case looks like rather typical example, where one should choose Symbian.