No all businesses do not experience the business cycle at the same time, but a recession is when many business are going through it at the same time. The progressive taxation system does not prevent downturns either. When we are in a recession, unemployment it higher and wages are lower. Lower wages means you pay less paxes in a progressive system, paying less taxes means you have more money to spend on other things that stimulates the economy. Yes the economy is like a roller coast, it is always going up and down (with an overall upward climb).
The government does control inflation. Its target is to keep it under ~3%/p.a. Without the plans they impliment, it would be growing exponentially. Look at Brazil, India, China etc. They economies are expanding at alarming rates, but their governments are not intervening to control inflation and there inflation rates are 10-15%/p.a.
Two fiscal policies the government has is spending and taxation. Lowering taxes and the government spending more will stimulate the economy and help end bring us to our potential output from he recessionary gap. Raising taxes and spending less will help to lower the inflationary gap bringing us back to our potential output. This is first year Keynesian economics.
The economy is sporatic and you cannot predict when a recession is going to occur. When there are signs of a recession, that is when parliament will start to debate how to best fix the problem. This takes time. Then they decide what to do, but it takes time for the effects to be felt by the economy. This is lag.