How to implement a DC/DC converter module into circuit
Regarding 

1: Yes! you do need a trim resistor to set the output voltage on this supply, given that you want 5VDC you should put a 287 ohm resistor connected from trim to ground as per figure 27 and 28

2: Definitely you need an output capacitor, for what i can see the supply has no output filter capacitor, so yeah, it's basically indispensable. I would put a 100nf and 22uf ceramic in parallel with the alternative 680 one you suggested, just be sure to choose ceramics with a voltage rating a few times greater than the output voltage (for more info look for voltage derating on class 2 ceramic capacitors)

3: yes you can leave it floating no problem

regarding the input capacitor you definitely need one if you wish to pass any compliance test, they mention multiple capacitors (big electrolytic, small ceramic and tantalum) because they resonate at different frequencies, hence the 100nf for example is better at decoupling higher harmonics than the electrolytic ones (same reason why i suggested you do the same at the output).

regarding the tantalum, i would avoid them, you can put a big ceramic one which will yield a very low esr and enough capacitance if dimensioned properly, plus you will reduce the amount of conflict minerals on your design and eliminate the chance of your tantalum blowing up when connected to low impedance sources

also you can put a ferrite bead in the input if you want to reduce emi even further but that's not really necessary