In the U.S. the states are the fundamental building blocks of the federal system and so, since senators represent states at the national level, they are pivotal in creating policy.
However I wonder if there are programs which are not administered through or by states, but directly by federal government, for example in case when there is a necessity to approve a law that no states want to approve. Are they?
Yes, indeed, there are many programs that the states have no part of. The 10th Amendment says that the states and the people retain the rights/powers that are not delegated to the federal government by the Constitution.
Some primary areas that are delegated would include defense, foreign policy, and pretty much the entire federal judiciary.
States, for example, may not send ambassadors to foreign countries, nor may they declare war.
Likewise, state courts have no jurisdiction in federal affairs or affairs that cross state boundaries.
Finally, many programs are administered at the federal level. From the spooky - the NSA, the CIA, to simple ones like the National Weather Service, and the Pell Grant Administration, the Pension Benefit Guarantee Corporation (PBGC), to even the powerful ones like the Federal Reserve (alibeit also a quasi-governmental organization), there are all sorts of programs run strictly out of DC.
To add to Affable Geek's excellent answer, another notable example is Immigration policy - this was a legal basis for US Supreme Court rejecting Arizona's recent immigration law (SB 1070, in ARIZONA ET AL. v. UNITED STATES).
A decision on removability requires a determination whether it is appropriate to allow a foreign national to continue living in the United States. Decisions of this nature touch on foreign relations and must be made with one voice.
A pretty detailed analysis of the decision can be found on Forbes or ScotusBlog, among other places.
