In New Zealand, the only country where I am a citizen, permanent resident noncitizens are allowed to vote in political elections. This to me seems reasonable, since they have made a commitment to the country, likely pay taxes, etc. However, it seems that NZ is an outlier internationally and this is very uncommon. The US, of course, does not operate this way. I would be unhappy at having to give up my NZ citizenship, but it seems that I would not have to, so at this stage I intend to become a US citizen as soon as legally allowed (there are a few years to go yet).
What is more interesting to me is the fact that NZ denies voting rights even to citizens if they have not physically been in the country in the past 3 years (this is the standard length of time between elections, although unlike the US, early elections are possible). This seems very sensible to me – why should I live overseas for decades and yet influence policies that do not affect me but may have a big impact on those living in the country? Some other countries have such restrictions, but most (all?) are much less strict than NZ. Interestingly, not only does the US allow expatriates to vote in US elections as long as they want, but in 38 states US citizens who have never lived in the US are allowed to vote! Of course, well under 2% of US citizens live abroad, whereas for NZ the figure is apparently more like 15-20%.
In the US, as in NZ and most countries worldwide, the age of voting is 18. The US has a long history of malicious attempts by authorities to restrict voters from exercising their rights, some of which have been tried very recently, and this seems to have caused some people to reject anything that reminds them of such practices. However I really don’t see why there is such strong opposition to some form of voter ID requirement. It is widely used worldwide (NZ does not have it) and according to opinion polls, is overall a popular idea among US voters. It doesn’t seem worth investing political capital into fighting against it on principle.
Apart from being underage or out of the country for too long, there are other ways to lose one’s right to vote. In NZ, prisoners serving sentences of less than 3 years may vote, but not those serving longer sentences, and there are restrictions for some mentally disabled people in secure facilities and for people recently convicted of corrupt electoral practices.. In the US, it varies by state (see this comprehensive guide) and typically requires conviction for a felony (by definition, a crime that is punishable by death or more than one year in prison). In Florida, for example, certain released felons have lost their right to vote permanently. This seems excessive, and at odds with the concept of paying a debt to society and being allowed to be rehabilitated.
Moving to restrictions on candidates, the Constitution places minimum age restrictions on candidates for federal elections: 35 for President, 30 (and 9 years of US citizenship) for Senate and 25 (plus 7 years of US citizenship) for the House of Representatives. For President, naturalized citizens are not eligible (so you won’t see me running in a few years). I can see why this was instituted, but it seems faintly ridiculous in the modern world. Would Arnold Schwarzenegger really not have acted in the best interests of the US if he had been allowed to be President? Interestingly, although being in prison cramps one’s style as a voter, in the US it is in general not a legal obstacle for candidates. The exception given by the 14th Amendment, Section 3 for those who commit insurrection is very topical this year, being explored in several cases including against Donald Trump (by the way, the disqualification can be overridden by a 2/3 majority in House and Senate). I am not aware of other legal restrictions on candidacy (although each state has rules for ballot access and may have other restrictions). By contrast, many other countries have other restrictions on presidential candidates (my favorite one so far is that Indonesian candidates must fear God Almighty; many Muslim-majority countries require the candidate be a Muslim, and the Maldives even requires the Sunni sect). NZ does not have a president, and to be a member of Parliament, a necessary condition is to qualify as a voter.
The job of President appears to be emotionally demanding. The rapidly graying hair of many recent presidents during their tenure is easily noticed. Typically the President is called on to make difficult decisions that no one else can (as Truman said, “the buck stops here“). A minimum level of physical fitness and stamina, and intellectual ability, is needed, but these should not be overstated. There have been fairly successful presidents who were obese, physically disabled, or poorly educated. Recently I have seen many people claiming that “X is unfit to be President” on the basis of being too old, too uneducated, too inexperienced, etc, and presumably many of these people would agree with instituting (if possible) more restrictions on who can be President. While it is reasonable for a voter to weight these personal characteristics when deciding whom to vote for, it is not reasonable to try to use them as general disqualifying criteria. It is not that I agree with Roman Hruska that mediocre people deserve representation, but I do think that every person has strengths and weaknesses, it is hard to measure all of them, and decisions on fitness for the job ultimately involve tradeoffs. Almost certainly, no candidate can combine all the strengths of Obama, Jefferson, Clinton, Biden, Truman, Lincoln and Eisenhower, for example. Whether someone is really too old, too uneducated, or too frail to serve ought to be decided by voters.