By Ju-min Park, Hyunjoo Jin, Cynthia Kim and Joyce Lee

SEOUL (Reuters) - From trade and diplomacy to markets and budgets, South Korea struggled to contain the fallout from the president's brief but shocking martial law attempt last week, just as the country navigates an uncertain future with its ally the United States.

Life largely went on without disturbance for South Koreans on Monday, but behind the scenes officials scrambled as President Yoon Suk Yeol's administration faces criminal investigations and impeachment attempts that cast doubt over his future in office.

While he survived a first impeachment vote on Saturday, the opposition has vowed to try again and Yoon's own party says it is formulating a plan for him to eventually step down.

The uncertainty over who is in charge of South Korea pushed the benchmark Kospi index down 2.8% on Monday, while the smaller Kosdaq index dropped more than 5%, the sharpest daily drop since August. The won weakened almost 1% against the dollar, hovering at levels last seen on Tuesday night when Yoon declared martial law.

The crisis is also raising concerns about the military's chain of command in the face of nuclear-armed North Korea, which has bolstered its weapons arsenal despite pressure from the South and its U.S. allies.

Key ministers have resigned and many agencies are in crisis control mode, at a time when South Korea needs to be preparing for the arrival of the new Trump administration in the United States, which is Seoul's key military ally and a major trading partner.

South Korea must be unremitting in its efforts to restore the trust of its partners and once again measure up to the expectations of the international community, Foreign Minister Cho Tae-yul told officials on Monday, noting the crisis comes at a time when threats are increasing, including from North Korea's deepening military ties with Russia.

"While this may take time, we must be steadfast and earnest as we devote our diplomatic efforts to rebuilding trust," he said.

Former trade minister Yeo Han-koo, who previously worked with the Trump and Biden administrations, said he was worried about a power vacuum in South Korea at a time when the presidential office and government should be working with industries to prepare responses to President-elect Donald Trump's plans, which will directly impact Korea's export-reliant economy and companies, including chipmakers and battery makers.

"Considering the tariff disruption expected under the Trump administration, strong government leadership and bold policy action are needed," he said. "But Korea is in turmoil."

South Korea also faced political upheaval with the impeachment and removal of then-President Park Geun-hye in a corruption scandal in 2016 and 2017 just as Trump was elected the first time, but the current chaos is worse, Yeo said.

"The uncertainty is a huge distraction," he said.

'DIPLOMATIC PARALYSIS'

On Monday the Justice Ministry approved a travel ban on Yoon, which would bar him from overseas trips including official summits with foreign leaders.

In order to sustain the country's semiconductor and export competitiveness, normal diplomacy must be supported, said Lee Jae-myung, leader of the main opposition Democratic Party.

"Diplomacy is in a state of complete paralysis," he told reporters. "If President Yoon does not resign immediately, it will cause irreparable damage to the exchange rate, the stock market, and the national competitiveness of the Republic of Korea."

Although existing talks on arms exports are ongoing, if political chaos is lengthened it may affect longer-term exports as government-to-government discussions are key in defence deals, defence industry sources said.

Hyundai Rotem said it was still in talks with Poland over the next phase of a massive arms deal including the company's K2 tanks, and it was doing its best to sign the deal this year, despite media reports that the arrangement could be in trouble.

Kyrgyzstan President Sadyr Japarov cancelled a planned visit to view the production site of South Korean utility helicopters during his visit to the country last week because of the turmoil, a defence industry source told Reuters, speaking on condition of anonymity.

Sweden's Prime Minister Ulf Kristersson, who had expressed interest in defence cooperation during talks with his South Korean counterpart in May according to South Korean officials, cancelled a planned visit after the botched martial law attempt.

U.S. Defense Secretary Lloyd Austin scrapped plans to travel to South Korea this week, though a senior State Department official was due to meet South Korean and Japanese counterparts for trilateral talks in Tokyo on Monday.

BUDGET FIGHT

Yoon cited opposition obstructionism over government budgets as one justification for his martial law decree. Now the fallout has stoked the row.

By law, parliament can only reduce government budget proposals - lawmakers need consent from the finance ministry to make any upward revision in government spending for the following year.

The opposition-controlled parliament earlier this month cut 4.1 trillion won from the Yoon government's proposed 677.4 trillion won ($473 billion) budget, putting parliament in a deadlock over spending.

By insisting on a smaller budget, the Democrats can curb Yoon's power over spending increases, which risks shrinking fiscal spending at a time when the economy barely skirted a technical recession in the third quarter.

Failure to approve the government's budget proposal before the end of this year would mean the government will turn to a provisional-budget system from January.

This would mean the government is allowed to go ahead with mandatory expenditures needed for the running and maintaining of government but it would face serious spending restraints as the finance ministry would not be able to issue more government bonds than in the previous year.

($1 = 1,432.9900 won)

(Reporting by Joyce Lee, Cynthia Kim, Hyunjoo Jin, and Ju-min Park; Writing by by Josh Smith; Editing by Gareth Jones and Ros Russell)