* Russia's first Winter Games open with 2-1/2 hour ceremony
* President Vladimir Putin hopes to prove doubters wrong
* Athletes, organisers look forward to full competition (Adds airline security, Google, ceremony details)
By Mike Collett-White
SOCHI, Russia, Feb 7 (Reuters) - Russian President Vladimir Putin attends the opening ceremony of the Sochi Winter Olympics on Friday determined to prove his doubters wrong after militant attacks, a row over gay rights and ballooning costs overshadowed preparations.
Putin has staked his reputation on hosting a safe and successful Games in the Black Sea resort of Sochi, where a spectacle before 40,000 spectators at the new Fisht Stadium will signal the start of the full sporting programme.
He will be joined by leaders from China, Japan and about 40 other countries in a show of support despite an international outcry over Russia's "gay propaganda" law passed last year, which critics say curtails the rights of homosexuals.
U.S. President Barack Obama, French President Francois Hollande, British Prime Minister David Cameron and German President Joachim Gauck are not attending the Games, and the U.S. delegation includes openly gay representatives.
Despite grumblings about poor accommodation and tight security, the mood among competitors and officials after a handful of early qualifying events in Sochi and at the mountain base 40 km (25 miles) to the northeast was upbeat.
"Conditions offered to the athletes are absolutely outstanding," said French Olympic Committee president Denis Masseglia on a clear, crisp morning.
Some 37,000 security personnel are on high alert over threats by Islamist militant groups based in the nearby north Caucasus region to attack the Feb. 7-23 Games, the most expensive ever staged at an estimated cost of $50 billion.
Separatist guerrillas seeking an independent Islamic state in Chechnya and neighbouring regions of southern Russia have vowed to disrupt the Olympics, which they say are taking place on land seized from Caucasus tribes in the 19th century.
Despite a "ring of steel" around venues, Russian forces fear a woman suspected of planning a suicide bombing may have slipped through.
Security analysts believe that an attack is in fact more likely elsewhere in Russia to humiliate Putin, who launched a war to crush a Chechen rebellion in 1999.
Twin suicide bombings killed at least 34 people in December in Volgograd, 400 miles (700 km) northeast of Sochi.
Moscow mayor Sergei Sobyanin on Friday tweeted to people in the capital "to be especially vigilant and careful on the metro, on ground transportation, at train stations etc." now that the Games were about to begin.
Underlining international unease over the threat of militant violence, the U.S. Transportation Security Administration is temporarily banning carry-on liquids, aerosols, gels and powders on flights between Russia and the United States.
The United States issued a warning on Wednesday to airports and some airlines flying to Russia for the Olympics to watch for toothpaste tubes that could hold ingredients to make a bomb on board a plane.
Deputy Prime Minister Dmitry Kozak told reporters in Sochi that Russian security services were working with colleagues from Europe and North America to minimise the risk of attack.
"There is no reason to believe that the level of danger in Sochi is greater than at any other point on the planet, be it Boston, London, New York or Washington," he said.
GOOGLE TARGETS "GAY PROPAGANDA" LAW
In addition to fears over security, Russia, hosting the Winter Games for the first time, has been under fire since passing legislation against promoting gay propaganda among minors. Putin says it protects young people, and has stressed that homosexuals would not face discrimination at the Olympics.
Google placed a rainbow version of its logo on its search page featuring the six colours on the gay pride flag - red, orange, yellow, green, blue and purple.
The page also includes a quote from the Olympic charter underlining the right to practise sport without discrimination.
Google Inc. declined to comment.
Konstantin Ernst, in charge of the opening ceremony, said Russian girl band t.A.T.u. would perform during the pre-show.
But he would not be drawn on whether Lena Katina and Yulia Volkova were chosen because they had shared a steamy kiss in the video to their 2002 hit "All the Things She Said".
The duo later announced they were not in a relationship, but have spoken out in support of the gay community in Russia.
Ernst gave little away about the content of the 2-1/2 hour show, although he said it would draw heavily on Russia's rich history of classical music rather than its less recognised pop scene, unlike the opening of the Summer Games in London in 2012.
He promised innovation, surprises and a performance of the Olympic anthem by renowned Russian opera singer Anna Netrebko. When asked how the Olympic flame would be lit, he replied in English: "This is the biggest secret ever."
Several competitors said they hoped the thrills and spills on ice and snow in Sochi and the nearby mountains would take centre stage once the Games were in full flow at the weekend.
"I don't really feel like the Olympics is a place for that kind of politics ... I think it's a place for sports and a place for cultures to put aside their differences and compete," said veteran U.S. Alpine skier Bode Miller.
Organisers have defended the costs of staging the Sochi Games, amid concern from Olympic officials that the huge price tag will put potential bidders off in future.
They said much of the infrastructure built for 2014 was designed to be used long after the Games finished, and the plan was to turn Sochi into a year-round resort, international sports centre and amusement park. (Additional reporting by Timothy Heritage in Sochi and Julien Pretot in Rosa Khutor; Writing by Mike Collett-White; Editing by Ossian Shine)