Why is periodicity seen in these certain properties? I missed my lesson on periodicity so had to teach myself, and have always forgotten to ask my teacher to explain to me why these trends are seen, which, unfortunately, the textbooks don't.

Density: the density of an element tends to increase across a period up to group 3, and then starts to decrease and become very low at group 5 or so. Why is this?

Melting and boiling point: they tend to rise up to about group 3 or 4, before falling rapidly after that. I would suggest that this is because group 4 elements tend to have giant covalent structures, so also a higher boiling point, whereas on the right hand side of a period they are simple molecular structures, so have a low boiling point. Am I correct?

Atomic size: atomic size always decreases along a period from left to right. My suggestion is that this is because there are more electrons and more protons as the group increases, so there becomes a stronger attraction between them, pulling the electrons closer to the nucleus. Am I right?

So the only one I have no idea about really is density, and the other two I just need clarification on.
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.