I never see States' representatives at international meetings or in any diplomatic relations. Are they responsible in any way for foreign policy?
Almost the entirety of foreign policy apparatus is laid out and reserved in Articles 1 and 2 of the Constitution, and therefore inherent in the Congress and the Presidency. For example the President has the power, with advice and consent of the Senate, to appoint Ambassadors, is Commander in Chief of the Armed Forces. Similarly the Congress can declare war. The treaty power is shared among them.
Article 4, section 4 of the Constitution does allow for this:
The United States shall guarantee to every state in this union a republican form of government, and shall protect each of them against invasion; and on application of the legislature, or of the executive (when the legislature cannot be convened) against domestic violence.
You can see it is limited in scope, and actually centralizes the foreign policy apparatus in the federal government in addition to Articles 1 and 2.
We also have the Tenth Amendment that reads:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
The Tenth Amendment allows states to engage in trade missions with countries. Here is a Massachusetts example. Note that limits on what they can and can't negotiate.
These sojourns certainly are not unusual: Cassey studied a decade-long period starting in the mid-1990s and found about 500 trade missions led by governors nationwide, with more than 40 states launching at least one.
“Governors tend to go to the places where there are already strong relationships,” Cassey said. Business leaders say it’s particularly important that the governor personally participate in the trips, rather than send emissaries.
A governor’s trade mission can close deals and open doors. These trips create deadlines to finalize a business deal or an expansion plan, to ensure the politicians have success stories to unveil during their tour. And a governor, as the state’s most prominent elected official, can line up in-person meetings with hard-to-get political officials and business executives.
Finally you have the Logan Act, which was established to prevent anyone from separately creating foreign policy other than the federal government. Wikipedia has a good enough summary:
The Logan Act (1 Stat. 613, 18 U.S.C. § 953, enacted January 30, 1799) is a United States federal law that forbids unauthorized citizens from negotiating with foreign governments having a dispute with the U.S. It was intended to prevent the undermining of the government's position. The Act was passed following George Logan's unauthorized negotiations with France in 1798, and was signed into law by President John Adams on January 30, 1799. The Act was last amended in 1994, and violation of the Logan Act is a felony.
To a large extent, states are explicitly banned from conducting most types of foreign relations by the U.S. Constitution, unless they get the consent of Congress first:
No state shall enter into any treaty, alliance, or confederation; grant letters of marque and reprisal; coin money; emit bills of credit; make anything but gold and silver coin a tender in payment of debts; pass any bill of attainder, ex post facto law, or law impairing the obligation of contracts, or grant any title of nobility.
No state shall, without the consent of the Congress, lay any imposts or duties on imports or exports, except what may be absolutely necessary for executing it's inspection laws: and the net produce of all duties and imposts, laid by any state on imports or exports, shall be for the use of the treasury of the United States; and all such laws shall be subject to the revision and control of the Congress.
No state shall, without the consent of Congress, lay any duty of tonnage, keep troops, or ships of war in time of peace, enter into any agreement or compact with another state, or with a foreign power, or engage in war, unless actually invaded, or in such imminent danger as will not admit of delay.
-- Article I, Section 10 of the Constitution of the United States of America (emphasis mine)
No, states defer to the federal government for foreign policy: defense, treaties, diplomats/embassies, etc. State governments may be involved with foreign businesses; for example, a state might negotiate with a business and offer tax incentives for the business to open an office or factory within the state and higher employees. But official foreign policy is a function of the federal government.
As a small-scale example, the reciprocal drivers license program Japan has is granted on a state-by-state basis, because driving laws and road tests, etc. differ from state to state.
Only countries have foreign policies.
As Google defines it,
a government's strategy in dealing with other nations.
A country's foreign policy, also called foreign relations or foreign affairs -policy, consists of self-interest strategies chosen by the state to safeguard its national interests and to achieve goals within its international relations milieu.
Since states are not nations and they still belong to the United States, they won't have their own foreign policy.
Imagine each state negotiate their own policy, it would result in a very divided country with no clear direction.
The only exception I can think of would be the Special Administration Regions of China which includes Hong Kong and Macau, however they still follow China's broad policy and direction.