Republicans in Congress suffered a humiliating series of setbacks on Tuesday on critical elements of their agenda, turning the Capitol into a den of dysfunction that has left several major issues, including U.S. military aid to Ukraine and Israel, in limbo amid political feuding.

As Republicans in the Senate torpedoed a border deal they had demanded, the bid by their counterparts in the House to impeach Alejandro N. Mayorkas, the homeland security secretary, collapsed amid Republican defections.

Then came one last bruising blow. Minutes after Republicans fell one vote short of impeaching Mr. Mayorkas — a punishment the party has promised its base ever since winning the majority — the House defeated legislation they put forward to send $17.6 billion in military assistance to Israel. The measure fell to opposition from Democrats who called it a cynical political ploy to undermine efforts to pass a broader foreign military aid bill including Ukraine. They were joined by a clutch of hard-right Republicans, who opposed the measure because the money was not paired with spending cuts.

Taken together, the events that unfolded on Capitol Hill on Tuesday offered a vivid portrait of congressional disarray instigated by Republicans, who are bent on opposing President Biden at every turn but lack a large enough majority or the unity to work their will.

They have sought to kill bipartisan efforts to send more military aid to Ukraine and to forge a compromise to secure the border against an influx of migrants, proposing instead to help Israel only and to push for the removal of Mr. Biden’s top immigration official. The back-to-back defeats on Tuesday showed that while they are adept at thwarting action on critical issues, they are hard-pressed to address any.

The paralysis left the fate of aid to Ukraine and Israel in peril, closing off what had been seen as the best remaining avenue on Capitol Hill for approval of critical military assistance to American allies. A broad measure that includes both is expected to fail in a Senate test vote on Wednesday, raising immediate questions about whether Congress could salvage the emergency aid package — and if so, how.