I landed US Aug, 1998. I began to work since 2001. I'm still holding F-1 visa, am I considered as nonresident alien?

When filing federal tax, can I use 1040NR?

I have a copy of IT203 for 2004 new york state tax, which was prepared by turbo tax software. The software put $0 on all new york state income, so I didn't pay any state nor new york city tax. In fact, I got all of them refunded. Is it true that F-1 students do not pay state nor city tax.
It's very important. Because I prepared 2005 state tax by myself, and I found that I owed NY state tax. so a check was sent to them.
best

You are NOT a non-resident alien. The exemption status granted under the F-1 visa is good only for five years. For sure, it expired on 31 Dec 2005. It may have expired at the end of 2002.

Hence, you MUST file Form 1040, 1040A or 1040EZ.

As for NY, if you earn income in NY, you must pay NY state income taxes. Same applies for NYC. The fact that you are in F-1 visa status seems to have no effect on that fact.