Why is periodicity seen in these certain properties?
For 2. I would add, on top of what you already have, that the elements start as metallic elements, hence we have metallic bonding. Across the period, the number of electrons increase, meaning there are more delocalised electrons among the metal cations so that electrostatic attraction increases along the period. For period 2, that's the trend.

For period 3 however, it is a bit more tricky because of the transition metal elements and I think Nicolau explains it better than I could in this answer. But I don't think that you will be required to know those for your level, but it's still interesting for your own knowledge. :)

For 3. I would add that the number of electron shells containing electrons remains constant. If it increased instead, the atomic radius would increase as well.

And you could use 2. and 3. to explain 1:

First, we observe that the atomic mass increases along the period, and that the atomic radius decreases. For giant structures, this generally means that the density increases ($\rho = \dfrac{m}{V}$ and while $m$ increases, $V$ is decreasing slightly, so $\rho$ increases)

When we hit the simple molecular structures, the density sharply falls again, because of the average distance between the atoms of the element has drastically risen, so the volume increases. With $m$ increasing gradually and a sharp rise in $V$, you get a sharp decrease in $\rho$.

These observations change down the groups though. For instance, Astatine is a solid, so you won't expect a huge fall in density across period 6 around group 5. But for the purpose of your syllabus coverage (I'm assuming High School), those should be enough.