Under what conditions (if any) should the government intervene in the market? Should the government provide public goods? If so, how would the government know how much of the public goods to provide? How could government intervention be constrained or controlled so that the rent-seeking and personal goals of policy-makers do not result in bad policy?

1. Under what conditions (if any) should the government intervene in the market?
According to John Keynes, the father of modern macroeconomics, a government should intervene when the aggregate demand is low and the economy is in disequilibrium. In other words, the government should only create policies that influence business decisions when free market forces have failed to stabilize the economy.
2. Should the government provide public goods? If so, how would the government know how much of the public goods to provide?
The government should provide public goods only if private firms cannot provide them. For example, in case there is a natural disaster that forces businesses to shut down, the government would be justified in providing public goods.
The number of public goods to be provided by the government would be determined by the demand. For example, if a county is submerged by floods, the government would have to provide shelter for all the residents of that county.
3. How could government intervention be constrained or controlled so that the rent-seeking and personal goals of policy-makers do not result in bad policy?
In order to stop the government from creating policies that don't benefit the general society, the public ought to be involved in the decision-making process. Before any law is passed or any public project is approved, the government should hold public debates to get the opinion of the citizens. When the public is involved, the public ensures that government intervention is for everyone's benefit.
https://www.investopedia.com/terms/k/keynesianeconomics.asp

https://www.investopedia.com/terms/r/rentseeking.asp

The debate over whether the government should intervene in the marketplace is a fierce one. There are people who believe that the government should rarely interfere in the marketplace. Other people believe there are times when the government should get involved in the marketplace.
There are a few instances when the government could be involved in the marketplace. One such instance would be when the economy is in a severe and prolonged depression. For example, the government got involved in the economy during the Great Depression because widespread unemployment and suffering existed. Many programs were developed to create jobs, such as the Works Progress Administration and the Civilian Conservation Corps, in order to get people back to work. There also were laws that were developed to reduce the likelihood that another significant depression would occur. For example, the Securities Act regulated the stock market, and the Glass-Steagall Act imposed new rules on banks.
Another example would be to provide public services or goods that the marketplace might fail to provide. For example, it would be difficult to have a private police force or a private military. The same is true for a national road system. In these cases, the government should step in and provide these services. The government would have to balance several factors when providing these services. One factor is how much these services cost. If so, does the government have the funds to provide this service? Once funding levels are determined, the governments would need to determine how to raise the money needed to provide these services. Generally, a cost-benefit analysis is done before the government commits to providing many of these services. After they are provided, the government also must consider the effectiveness of these services. For example, if crime rates were high, the government would have to consider increasing the number of police officers in the police force.
Government intervention and control can be constrained by creating boards or commissions to oversee the role of the government. For example, an independent police commission would oversee and review the work and activities of the police department. This commission could order the police department to change policies that were deemed ineffective and order the police department to implement new ones. Oftentimes, the media will play a watchdog role. The media will do reports on various projects in which the government is involved. These reports may expose improper actions by government officials, which may lead to changes in policies or activities.
https://www.economicshelp.org/blog/5735/economics/should-the-government-intervene-in-the-economy/