Does this mean I could omit a current limiting resistor as the internal impedance would limit it to 3.3mA (5v/1500Ω=3.3mA)?
Correct. If you want to be extra-safe*, you can use one and still get enough current/voltage to trigger the SSR.
What does "Zero-crossing (resistive loads) or random-fire (inductive loads) output" mean?
The zero-crossing version switches-on shortly after the AC power crosses-through zero. I think the advantage is that you don't get noise on the AC line if you switch at zero. SCRs and TRIACS always
at the zero-crossing. (It won't work as a dimmer, which needs to switch-on at different points/angles depending, on the dimmer setting.)
I'm not 100% sure why the non-zero crossing version is needed for inductive loads... so I'd better not guess.
I have also been reading that a snubber is not needed for SSRs. Will someone confirm or give argument otherwise?
On the input side? That's right. There is no inductive coil with a solid-state relay.
Finally, is something like this safe to control directly from an output pin or should I use a transistor to switch it?
With the low voltage & curent requirements, you can drive it directly.
* Let's say you've got your low-voltage contorller in one box and your high-voltage and SSR in another box with connectors & cables running in-between. If there's a limiting resistor inside your low-voltage box, the Arduino won't get killed if someone shorts-out the outputs.