Are there any limits to his power from Congress while he controls a majority there?
In theory (according to the Philippine constitution), the Legislative branch should provide a check on the President's spending and the Judicial provides a check on Presidential appointments (having to refer people to the President).
Likewise, the Legislative makes the laws, the Executive faithfully executes the laws, and the Judiciary compares the laws to the Constitution. This is very similar to the USA, because the USA was one of the inspirations for the Philippine government.
While the Philippines (thankfully) don't have the problem of having allied people in positions of power in all three branches like the USA does, it does have a similarity in that many of the standards of behavior were not codified to allow for greater flexibility, trusting that shame and respect for precedent would keep everyone in line.
However, the Legislative branch sought to impeach a head Judge not for malfeasance, but for a controversial ruling. Eventually, Duterte was outed as being a backer of this impeachment. In addition, both the Legislative and Judicial branches have involved themselves in jailing and freeing local officials accused of graft. To add to this mess, the Executive branch (Duterte) is making demands of the other two branches of government to further his political aims, and uses his bully pulpit to attempt to sway public opinion to his side.
Under the Constitution, it would take one third of the House of Representatives to initiate impeachment proceedings, then it is sent to the Senate for the final decision. Unfortunately, as long as the Legislature is allied with Duterte, he cannot be removed in this way.
Yes. According to the congress' Wikipedia page the congress has the following powers (I only quote the ones that seem like checks on the president):
Confirming presidential appointees through the Commission on Appointments;
To concur and approve amnesty declared by the President of the Philippines;
To authorize the President of the Philippines to fix tariff rates, quotas, and dues;
To authorize the President of the Philippines to formulate rules and regulations in times of emergency;
Under the voting requirements heading on that same Wikipedia page, it also says the congress has the ability to override a presidential veto. That does, however, require a two-thirds majority. The page is unclear if that requires such a majority in the house and the senate or in merely one of them.