The World Cup squads are preparing to fly, the performers for the opening ceremony are in rehearsal, and Fifa officials have finally taken the reins for the run-in to Brazil 2014.

But with less than three weeks until the competition kicks off on 12 June, the venue for the first match has still to complete trials that should have been finished months ago.

The delays at São Paulo's Itaquerao stadium – where the roof had to be scaled back at the last minute – are among several remaining hiccups in the preparations for a tournament that has been dogged by the deaths of construction workers, protests, strikes and overspending.

Government officials also warned this week of potential problems with mobile phone coverage during matches, while Fifa executives – who have moved to take control of the stadiums in the past few days – said there was still outstanding work to complete to ensure the event was a success.

After a final inspection of the 12 venues, the Fifa secretary general, Jerome Valcke, gave the tournament the green light, but admitted preparations were not yet complete. "The Fifa World Cup has already arrived in Brazil. And the whole world is watching in anticipation," he wrote on Twitter. "We have busy days ahead of us with still a lot to be done."

The most pressing concern is the Itaquerao stadium, which will stage the opening ceremony. Jennifer Lopez, Pitbull and massed ranks of samba dancers, capoeira performers and Salvadorean drummers are due to entertain the 70,000 crowd before the the first match between Brazil and Croatia.

But construction of the venue – which was supposed to have been finished a year ago – has continued up until the last moment and the facilities have yet to be tested with a full attendance.

Fifa normally insists on three trial events before a World Cup, but the Itaquerao has only had one – a domestic league match between Corinthians and Figueirense last Sunday that drew about 36,000 people.

At that event, many fans were forced to seek cover during a hail storm because glass roofing panels will not arrive until after the World Cup. Although this does not affect players on the pitch, it could cause problems if there are storms during the tournament.

Additional seating that will be used during the World Cup was not yet in place, metal detectors were not operational and mobile phone coverage was poor. To rectify these problems, a second test event has been scheduled for 1 June, 11 days before the start of the World Cup.

"It is vital for us that all facilities will be tested under full match conditions including the temporary seats and associated facilities," Valcke said.

Two smaller stadiums that were also delayed – at Curitiba and Cuiaba – have been given the go-ahead after only one test event, though there is still considerable work to be done on the stadium exteriors and transport links.

At the test event for the Arena Pantanal in Cuiaba last week, there were no parking spots for the team buses and domestic journalists complained of bad conditions in the toilets and press centres. A light railway that was supposed to ferry supporters to the matches will not be complete until next year.

Fifa has also warned that construction of temporary structures for broadcasters in the stadium at Porto Alegre is also behind schedule.

A broader concern is the weakness of telecommunications networks, which led to mobile phone blackouts during last year's Confederations Cup. Technicians complain they have not been given adequate time to upgrade networks at stadiums, airports and other areas where data traffic is likely to surge during the tournament.

The communications minister, Paulo Bernardo, warned of likely telecoms and internet problems at half of the 12 World Cup stadiums, including Itaquerao. Fifa's Valcke said urgent action was needed: "We don't want Brazil to be remembered as the worst World Cup of all time because the journalists could not get their stories out to the rest of the world," he said.

The often shambolic preparations have added to public frustration at the expense of the tournament, which – at about $11bn (£6.5bn) for stadium and infrastructure construction – is about triple the cost of the South Africa World Cup. It is also far over budget. Earlier this month, the government acknowledged the cost of the Mané Garrincha stadium in Brasilia has almost tripled to $900m. Official auditors are investigating allegations of price-gouging and corruption at this stadium and more than a dozen other cases related to construction of venues.

A major share of the construction funds have come from the public purse despite earlier government promises "not to spend a penny of public money" on stadiums. Fifa meanwhile is expecting record revenues from broadcast rights, sponsorship deals and ticket sales.

Protests against the tournament continued on Thursday night when a rally of about 20,000 demonstrators marched through the centre of São Paulo. The city's always congested traffic was also hit by a bus workers' strike. Police in 14 other states also went on strike, adding to pre-tournament social unease.

These glitches can still be ironed out before the tournament, but time is running out. Australia will be the first of the World Cup squads to arrive next week and others will follow soon after.

The delays and overspending have prompted criticism even from Pelé, the former national team hero who is usually the most vocal supporter of the football establishment.

"It's a pity that we still have a stadium that's not yet finished,' Pelé said earlier this week in reference to the missing section of the roof at the Itaquerao. "It's clear that politically speaking, the money spent to build the stadiums was a lot, and in some cases was more than it should have been."