When new BP boss Bob Dudley slides into the seat freshly vacated by Tony Hayward next Friday he faces a huge task – resurrecting a brand that has been savaged by the gulf oil-spill disaster.

The softly spoken American, the first non-Brit to hold the chief executive job, is a long-standing BP man so might not be overly fazed by the scale of the task. But he will be acutely aware that the oil giant has a track record as a career-killer: three out of four predecessors – John Browne and Bob Horton as well as the hapless Hayward – have been forced out under a cloud. It is a big job, but Dudley's five priorities are clear:

Schmooze the US

Once the most swaggering energy enterprise, BP has been brought low by the Gulf of Mexico well blowout and by the Texas City refinery fire, Alaskan pipeline spills and propylene trading probes. All these dismal events have happened in America, where roughly 25% of the production and 40% of the company's profits are made and where BP's name has been most devastatingly trampled by politicians and public alike.

Dudley is as well positioned as anyone – and certainly better placed than Slough-born Hayward – to do this, not least because he speaks with a familiar southern-state inflection that plays well with the home crowd.

He is steeped in US oil culture, from his earlier career with Amoco, which was later bought by BP, and from spending the last few months in charge of BP's continuing effort to clean up the beaches since the Deepwater Horizon explosion on 20 April.

The 55-year-old will need to make a huge effort to convince Americans that he is going through all the local rigs, refineries and depots with a fine-tooth comb to weed out any practices that could suggest safety is not of the highest degree.

He will have to overcome the in built scepticism of the locals who remember when Hayward promised to employ "laser-like focus" on the same issue in a bid to move on from the dark days of the Texas City fire, when 15 workers were killed and 170 injured.

Dudley will no doubt want to strengthen the corporate structures of BP in the US, but his ability to make major progress will be hindered by the raft of official investigations going on into the Macondo well.

BP has published its own report and attempted to deflect much of the blame on to other operators but will be hoping that more independent probes do not pin a "gross negligence" charge on the company.

Dudley will have to put a lot of time into fighting some of the current legal claims but gross negligence would open the floodgates to a much larger wave of court actions that could yet sink BP.

Give investors a new vision

The company was the biggest in US deep-water exploration and production but its future there as an operator remains in doubt. There is still a drilling ban on new wells but when that is lifted in November it remains uncertain how BP's applications for work permits will be received.

Would it be better for BP to cut its losses in the US, sell up and ship out? The likes of ExxonMobil and many others would relish the chance to pick up choice assets. Alternatively, BP could take a back seat and concentrate on being a stakeholder not an operator. That has not been the company's style hitherto.

BP likes to say privately that its brand remains untarnished and its prospects in deep-water areas, such as Angola and Brazil, remain very positive. But the decision to withdraw from the Greenland offshore licensing round shows that the company is still considered a liability, at least in very environmentally sensitive areas such as the Arctic.

Despite all this, Dudley is likely to push the idea that BP can still play to its strengths as an expert in deep-water drilling, arguing that its experience in the Gulf of Mexico leaves it able to deal with any future crisis better than anyone.

BP has already committed itself to selling $30bn of assets to bolster the $20bn it has been forced by the US government to set aside to pay for the liabilities emanating from the Macondo well spill.

Dudley is likely to argue that this fits into a wider shrink-to-grow strategy that has been encouraged by investment banks such as Morgan Stanley and was already underway prior to the gulf spill. Like many of its peers, BP was rapidly retreating from some "downstream" areas such as refineries and gas stations. Should it sell them off completely or split the group into two, even while concentrating on value not volume?

Go to Russia with love

BP is heavily dependent on its relationship with Russia, yet while Hayward was public enemy number one in America, Dudley was held in almost as low esteem in Moscow after a punch up between BP and his Russian shareholders inside the joint venture TNK-BP.

Dudley, who ended up being forced to flee the country, hopes that putting Hayward on the board of TNK-BP as a non-executive of TNK-BP will help cement a business that provides a quarter of the wider BP group's production.

The Briton may have pariah status in Washington, but the BP board reckons that Hayward still has a good dialogue with the Kremlin after he allowed the Russians to put their own representatives into the top jobs at TNK-BP. Hayward's gain is the loss of Andy Inglis, BP's head of exploration and production, who is stepping down from the TNK post.

Weed out the Swede?

Dudley will need to decide whether Carl-Henric Svanberg's continuing presence as BP's chairman is going to undermine his planned recovery operation for the group. It is not obviously in the American's gift to dispose of his Swedish chairman, but there are clearly ways in which chief executives can use their influence to trigger board action.

Many outsiders believe that Svanberg is lucky to have survived when Hayward did not. Even in his home country, Svanberg was widely criticised for keeping a relatively low profile when the flak was flying in Washington and failing to be publicly supportive enough of his beleaguered chief executive.

Maybe others, such as Inglis, might be moved aside as part of a new broom attempt to separate the new BP from the old pre-gulf company. Already there are whispers that Dudley is looking for a new head of communications who would outrank current PR boss and former Financial Times editor Andrew Gowers.

Decide the dividend

Dudley must decide whether to start paying dividends again after they were suspended by Hayward under pressure from Barack Obama. Parts of the City are desperate to see him do this, perhaps as early as next spring, but some say it would not make sense in public relations terms to start rewarding shareholders so soon after the disaster and when the beaches are still being cleaned.

Dudley will factor this in to other wider questions about company strategy – he may prefer to channel the cash into more exploration and development – or even acquisitions, which would almost certainly be outside of America.

But returning the payouts to shareholders would be laden with symbolism. It would show BP back on the road to recovery while separating Dudley from Hayward and Horton – who both lost their jobs after being forced to axe the dividend.