Court opens for wide-ranging docket despite government shutdown and will sit at least through the end of this week

The supreme court began its new term on Monday by turning away hundreds of appeals, including Virginia's bid to revive its anti-sodomy law.

The justices took the bench just past 10 o'clock, even as much of the rest of the government was coping with a partial shutdown. Chief justice John Roberts formally opened the new term without any reference to the partisan impasse over the budget and the new healthcare law that his vote helped uphold in 2012. The court has announced it will operate normally at least through the end of this week. The justices are hearing six arguments, including a challenge to limits on campaign contributions.

Among the appeals denied on Monday was Virginia attorney general Ken Cuccinelli's request to review a federal appeals court ruling that threw out the state's ban on oral and anal sex. Ten years ago, the supreme court struck down a Texas anti-sodomy law in a case involving two adults. Virginia argued that the Texas ruling did not apply to sex acts between adults and minors. The justices did not comment in rejecting that argument.

The court also declined to hear, at least for now, Argentina's appeal of a ruling that orders it to pay hedge funds that bought up some of the country's unpaid debt from its default in 2001. The country is continuing to pursue its case in federal court in New York and could file another appeal with the supreme court.

The new term may be short on the sort of high-profile battles over healthcare and gay marriage that marked the past two years, but the court has agreed to hear important cases about campaign contributions, housing discrimination, government-sanctioned prayer and the president's recess appointments. Abortion, contraceptive coverage under the new healthcare law and cellphone privacy also may find their way onto the court's calendar.

Several of those cases ask the court to overrule prior decisions – which would be bold actions in an institution that relies on the power of precedent.

"There are an unusual number of cases going right to hot-button cultural issues and aggressive briefing on the conservative side asking precedents to be overruled," said Pamela Harris, a Georgetown University law professor who served in President Barack Obama's Justice Department.

Paul Clement, a frequent advocate before the court and the top supreme court lawyer under President George W Bush, agreed that the opportunity exists for dramatic, precedent-busting decisions. But Clement said each case also offers the court "an off-ramp", a narrower outcome that may be more in keeping with Roberts' stated desire for incremental decision-making that bridges the court's ideological divide.

The campaign finance argument on Tuesday is the first major case on the calendar. A 5-4 decision in the Citizens United case in 2010 allowed corporations and labor unions to spend unlimited sums in support of or in opposition to candidates, as long as the spending is independent of the candidates.

The new case, McCutcheon v Federal Election Commission, is a challenge to the overall limits on what an individual may give to candidates, political parties and political action committees in a two-year federal election cycle – currently $48,600 to candidates and $123,600 in total. The $2,600 limit on contributions to a candidate is not at issue.

Since the Buckley v Valeo decision in 1976, the court has looked more favourably on contribution limits than on spending restrictions, because of the potential for corruption in large contributions. The big issue in the current case is whether the justices will be just as skeptical of limits on contributing as on spending. Three justices – Anthony Kennedy, Antonin Scalia and Clarence Thomas – have signaled their willingness to do so. It remains to be seen whether Roberts and Samuel Alito, the other two members of the Citizens United majority, are willing to go along.

The following are among other top cases already set for review:

• Greece, New York, a suburb of Rochester, is asking the court to uphold its practice of opening town council meetings with a prayer, despite an appeals court ruling that found the invocations to be a violation of the First Amendment because they almost always were Christian prayers. The court could use the case to rule that courts should take a more hands-off approach to religion in the public square, or it could hold more narrowly that the town's practice is consistent with a 1983 decision upholding prayer at the start of government meetings.

• Mount Holly, New Jersey, is defending a plan to demolish and redevelop a rundown neighborhood against claims that it discriminates because it disproportionately affects African-American and Latino residents. At issue is whether there also must be an intent to discriminate under federal housing law. The issue affects a range of transactions involving real estate and applies to banks and mortgage companies as well as governments, such as the one involved in this case.

• Michigan is fighting to preserve a constitutional amendment that bans the use of racial preferences in education after a federal appeals court ruled that the constitutional ban is itself discriminatory. This case, unlike last term's look at a University of Texas admissions plan, does not involve the viability of affirmative action, but rather whether opponents of racial preferences can enshrine that ban in the state constitution.

• Massachusetts is defending a law that creates a 35ft buffer zone at abortion clinics, in order to limit protesters' ability to interact with patients. The court upheld a buffer zone law in Colorado in 2000, but Roberts and Alito have replaced members of that majority and are considered more sympathetic to the free-speech claims of the protesters.

While several cases call into question high-court precedents, the justices will be writing on a blank slate when they take up the president's recess appointment power under the Constitution. In that case, the court will confront an appeals court ruling that would effectively end the president's ability to make such appointments, if it is left standing.

Former Justice Department official Peter Keisler said justices often ask a lawyer for the best case in support of his argument. "No one is going to ask that question because [there are] none. No Supreme Court decisions are material here," said Keisler, a partner at the Sidley, Austin law firm in Washington.

The impasse that led Obama to install members of the National Labor Relations Board and Consumer Financial Protection Bureau director Richard Cordray in office without Senate confirmation has been resolved. So what remains of the issue is whether Obama and his successors will be constrained in the future.

The topic splits Democrats and Republicans, but their view of the matter is almost entirely dependent on which party controls the White House.