Is it possible for a sun-like star to hold on to about 16 planets? If so, where could the habitable zone be located at?
TL;DR: Yes, you can have those, if you are willing to use a loose definition of "planet". Or you can keep it under/around 10 and meet the IAU expectations (hard science). As for the habitable zone, depends on your chosen star type.
But it is very hard to actually pinpoint the number of planets in a system.
The system with the most planets we know so far is HD1080 that has between seven and nine planets. We know that Sol has eight planets and a ton of planetoids (sedna, pluto, etc) so that is a good ballpoint number for "hey that's a lot of planets".
Maybe we could have one more planet between Mars and Jupiter if all the mass in the asteroid belt would congeal into a single body. Maybe one more if Pluto gathered the mass of the Kuiper belt. But this is speculation. More on planetary mass
Sixteen may be too much. Who knows. You could get away with that, but the star would have to be way bigger than the sun. Because of Kepler's third law, the orbits of the planets (calculate here) would be influenced by the star, and further planets would need a bigger gravitational pull to remain in orbit. And there are tons of problems with multi-orbit like resonance
As a P.S: don't throw dwarf planets in the mix (sorry Pluto) or you will get over 100 just in our solar system. (thanks Phiteros for pointing that out). Or do throw those in, just to meet your quota.
On 16 planets, yes it is possible, if Jupiter were smaller and didn't interfere with the asteroid belt you'd likely have another planet in that orbit. Though not easy it is possible that 7 more planets could be captured in the kuiper belt and clear their regions, but most would likely need to be captured from outside our solar system which would be a challenge. One or two yes, seven would not be common.
On the habitable zone, if your system were like our solar system then it can be argued we only have Earth as a habitable planet. But I'd argue if Mars were only a little bigger and had a substantive magnetic field it would be habitable too. Now if you really want to maximize the number of habitable planets you have to get creative. So instead of Earth we replace it with a heavy Jupiter, with 4 large moons in its orbit, all now within the habitable zone. Next replace Saturn or Uranus with a red dwarf star, essentially creating a binary star system which is actually quite common. Then the red dwarf can have at least one habitable planet in its orbit, again if you are creative you might squeeze two by having a super Earth with a super moon ie both 2.5 times bigger than our Earth and moon.
The habitable zone depends on the star, not on the planets.
Cleared this, in principle it is possible that more than 10 planets orbit a star. The "trick" is in preventing the formation of gas giants like Jupiter and Saturn which, with their mass, account for most of the mass of the system (central star excluded) and also whipe the surrounding from other bodies. Mass-wise you could split the mass of Jupiter in more Mercury-sized planets and easily boost the planet count for your system.
The problem is that, material-wise, rocky planets rely on metals for their formation, and metals are rare in the space, while there is plenty of gases to form gas giants and you cannot have earth sized gas planets.
Since our knowledge of planetary system is pretty limited and surely not yet statistically significative, we cannot rule out such possibility.
Sixten? Only sixteen?
Theoretically, a star can have up to four hundred and sixteen platens in the habitable zone. See here
What's your time scale? A young star system early in its lifespan could have a few "extra" planets and proto-planets in meta-stable orbits. Relatively soon in geological timeframes, orbital resonances are going to stack up and pull things apart, but for the present everything is behaving itself. So the planets aren't going to crash into one another immediately, but I wouldn't set up shop there for more than a couple thousand years. And asteroid activity is still very high, so mind your flight vectors and keep particle shielding at full power at all times.
Of course, you're probably not going to see native lifeforms in a system like that, but some enterprising settlers might set up shop for research/economic purposes.