Very generally, federal agencies write nation-wide minimum sets of regulations (which have to be based on laws passed by Congress, which in turn have to be grounded in allowances in the Constitution - powers not explicitly granted to the federal government belong to the people/states [see below]). Some of these federal agencies have enforcement officers, but nowhere near enough to police everything. In the case of the EPA, the rules relate to pollution, toxic materials, etc (lead paint, asbestos, petro-chemicals, etc). FDA is about food production and drug safety. OSHA is work-place safety.
States (and cities, counties, etc) generally have their own building codes and enforcement agencies. But, the states still have to follow federal rules where they exist.
So, in this case, there are EPA rules about lead paint handling. Very likely three are also OSHA rules dealing with worker safety when dealing with lead (wearing a respirator). And while the EPA might have some enforcement officers, they won't be interested in a single house renovation (more likely interested in factories, sky-scrapers, etc). OSHA also has enforcement, but a lot of that ends up being after-the-fact (ie somebody loses a limb, OSHA gets called, and shuts down a factory until remedial action taken).
But, the local building inspectors office is able to enforce the EPA rules as well as local building codes. And there's probably a local work-place safety office as well. But, in the case of small construction jobs, local code enforcement is the easiest/fastest way to get something on-site changed.
With respect to the Trump administration, you are correct - he went out of his way to install department heads who were either incompetent or had business goals opposed to the agency they were tasked with leading.
[] - The 14th Amendment gives the federal government the power to ensure all people are protected equally under the law. Pretty darn broad power there. Similarly, the Commerce Clause in the Constitution grants the federal government the power to regulate trade between the states and other countries. Again, you can read that very widely - pretty much any serious level of commerce impacts more than one state.