* 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 (updates with ceremony starting)
By Mike Collett-White
SOCHI, Russia, Feb 7 (Reuters) - Russia's Winter Olympics officially got underway on the Black Sea coast on Friday with a burst of fireworks and a dazzling opening ceremony that President Vladimir Putin hoped would dispel fears of militant attacks and a row over gay rights.
Putin has staked his reputation on hosting a safe and successful Games in the resort town of Sochi, where a high-octane show before 40,000 spectators at the gleaming new Fisht Stadium signalled the start of the full sporting programme.
He was 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 day.
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 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, gave little away about the content of the 2-1/2 opening show, although he said it would draw heavily on Russia's rich heritage and take viewers on a tour of the host nation's history.
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."
He would have been disappointed, however, by what appeared to be an embarrassing technical glitch right at the start when a light show depicting the five Olympic rings instead showed only four, with looked like a sunflower in its place.
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. Editing by Mitch Phillips)