First I will acknowledge it is difficult to write on this topic neutrally, even 150 years later, as the scar of the Civil War still runs through the country. I'll do my best to remain factual.

Did the Southern States make any attempt to secede from the Union, prior to 1861, through an act of Congress?

I cannot find any record of a serious attempt, no.

If not, what diplomatic/peaceful means were attempted?

The long running battle of States Rights and the idea of Nullification could be seen as this. At its head is the argument over Compact Theory, which has been rejected multiple times by the Supreme Court before and after the Civil War.

At the heart of the Nullification and Secession legal argument is Compact Theory, the idea that the US Federal Constitution is a compact between the states and any state can withdraw at any time. The foundations of the argument in favor are the Tenth Amendment which states that

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 argument against is based on the idea that the Constitution is a document between the people and the Federal government, the states are not a party. This relies on the preamble which says "We the People" not "We the States".

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.

Furthermore, it is "We the People of the United States", acting as a body, not the people of each state deciding on their own. Various attempts by states to show local majority support for nullification relied on Compact Theory to be considered valid.

Compact Theory was ultimately shot down by the Supreme Court long before the Civil War. Chisholm v. Georgia (1793) upheld that Federal courts could hear disputes between states (immediately voided by the 11th amendment). Martin v. Hunter's Lessee (1816) ruled that states did not have the power to re-interpret Federal treaties. McCulloch v. Maryland (1819) upheld Federal authority to create a bank. All rejected arguments based on Compact Theory, but its use as a justification for nullification and secession persisted.

This came to a head in The Nullification Crisis of 1832. Had it not been resolved diplomatically, it could have lead to secession.

Mostly southern states, losing influence in the House of Representatives due to a declining share of the population, tried a new tactic. They argued they could selectively ignore Federal laws which their state felt were in violation of the Federal Constitution. This argument bypassed the Constitutional authority of the Supreme Court to make that decision. This all came to a head in 1832 over federal tariffs which the South did not like since they depended so much on cotton trade with Great Britain.

The rhetoric got hot and South Carolina passed legislation declaring the tariffs null and void within its borders on February 1st, 1863. Militias was raised to defend against Federal troops. They had a right to be afraid, the threat of Federal troops had been used against states to enforce Federal law before. Troops threatened New England states which refused to provide men for the War of 1812. Before that, the Whiskey Rebellion of 1791 was put down.

In the end, Congress negotiated a compromise tariff which satisfied both sides enough to back down. The particular issue was resolved, but the basic problem remained.

Similar negotiated issues which may have lead to secession include the Compromise of 1850 and the disastrous Kansas-Nebraska Act.

What about various attempts by the southern states to poll their own people? While this may be peaceful, it is not diplomacy. As explained earlier in discussion Compact Theory, the Federal government rejected the idea that a state, or even its people, could decide alone to secede. Declaring you're going to ignore Federal decisions is not diplomacy, it is the end of diplomacy.