First question: In presidential systems like the US, does the president (as part of the executive and not the legislature) make any laws? E.g. how did the Trump administration enact the ‘travel ban’ and the 2017 tax cuts?

Answer: The short answer here is: no, the president does not make any laws. In practice, if they want legislation to be passed then they must find a friendly representative (usually from their own party) to introduce it into the legislature. Even then, though, it can be hard to pass legislation because party discipline is often weaker in presidential systems (because the president is outside the legislature and cannot promote representatives, or threaten an early election). This means that, often, the support coalition for legislation must be built on a case-by-case basis, and representatives can extract concessions for their support (meaning that the legislation often gets altered from the president’s ideal form before it is passed). With the preceding in mind, the ‘travel ban’ (I assume you mean in 2017 rather than in relation to coronavirus) and 2017 tax cuts might seem inexplicable. However, President Trump used an executive order to pass the former; executive orders are a constitutionally defined power of presidents to direct the practice of government departments, and do not constitute legislation. In fact, they can be overturned by legislation passed in the legislature (Congress, in the case of the US), and are subject to judicial review as well. As for the tax cuts, they were passed as legislation by the Republican Party in Congress, and President Trump had to rely on his allies in the party to introduce the legislation and ensure that it passed through both chambers.

Second question: In parliamentary systems like the UK, what powers do the Prime Minister and the Government have besides voting in parliament (as opposition MPs also do)?

Answer: Some very important ones: the power to introduce legislation, the power to call an election and, in the case of the PM, the discretion to promote or demote MPs to and from ministerial positions. In the first of those cases, most of the time legislation is introduced in a parliamentary system it is done so by the government, so they act as the legislative agenda-setter. In the second case, the PM and government usually have the ability to call an election (fixed parliamentary terms in the UK are a recent innovation and may be removed under this government), which of course can be used to ensure that their MPs vote in favour of their legislation (because many MPs would rather avoid fighting early elections). And finally, in the third case, the PM (and the party whips who support them and act as enforcers) can use promotion or demotion to ensure loyalty amongst MPs. So, all of these powers function through the parliament, which shows the nominal importance of the institution, and hence the reason for calling it a parliamentary system (though as Professor Hix pointed out, the power lies with the government!). Outside of parliament, the government, of course, also has some discretion over how government departments are run and how policy is implemented, or in other words the power to direct the day-to-day functioning of government.

Third question: Presidential systems are said to have more veto players than parliamentary systems. Is that only because the executive and the legislature are mutually independent, or are there other reasons (other veto players)?

Answer: Yes, the most important additional veto player is the president themself, who has a constitutionally defined veto; such a powerful additional block on legislation generally does not exist in parliamentary systems (as you say, this relates to the mutual independence of the executive and legislature in presidential systems). Additionally, presidential systems are often set up to have two powerful chambers in the legislature, whereas parliamentary systems may be unicameral or have a weaker upper chamber (as is the case in the UK). Additionally, there are then other things that may be associated with presidential systems, such as federalism, that introduce veto players. Of course, such things are not exclusive to presidential systems (e.g. Germany is a federal parliamentary system), so the association may emerge from historical or cultural context (e.g. countries in Latin American that adopted constitutions similar to, or influenced by, that of the US, thus implementing presidentialism and federalism) and thus are not integral to presidentialism.