But he's not tackling corruption, even police officers of high rank are corrupt. That was not so prevalent during the reign of Saddam. He's done a good job with diplomacy playing the game with the west, but has not got strong rule.
Yes, but that is the fault of the political system. Al-Abadi's hands are tied as long as the current system is in place. He knows that he cannot confront the usual suspects yet this soon after defeating the ISIS cancer and with monkeys in the North conspiring (sabotaging openly in fact) and now protests. Also the economy is a challenge although I believe that it will soon improve and the potential is enormous to say the least.
Who is a better choice currently? Al-Abadi enjoys widespread support from non-Shia Islamists. Al-Sadr is apparently an "nationalist" but we all know about his ideology. He is after all an cleric first and foremost. Not a politician. Then there is Al-Amiri. From bad to worse.
Al-Maliki (cretin) ran for office again.
Al-Abadi is a well-educated individual that speaks English and has done a good job overall (compared to the circumstances before and challenges).
I have no idea who could do better? Have in mind that there are better politicians out there (probably) but will they have a chance to win? The only politicians that can win are those who are members of the leading political parties. Others have no chance even in coalitions.
Of course there is also the option of a military coup but I don't see that happening and the world powers will most likely be against it (USA in particular) but this is hard to tell.
Obviously corruption is an obvious issue that must be tackled sooner rather than later (before sentiments boil over in the society and the divide between well off people and not so well off people grows much larger) but unfortunately this is common in the region as a whole. Some countries are doing better than others but this is a big problem overall.
https://en.wikipedia.org/wiki/Corruption_Perceptions_Index
A mentality change is needed among average people and for systems (political but not only) to reform.