Electrical – Are these wires dangerous

Electrical hum like that can be a symptom of a loose connection in the wiring. It's possible that something came loose when the wires were pushed up into the junction box. If so, the worst case would be an electrical fire, so it's definitely something to have looked at. Less terribly, you could lose power to the fan, where you'd probably have the electrician out again anyway.

If you're comfortable looking into it yourself, turn off power to that circuit at the service panel, open the cover and look at each of the connections. Your profile doesn't say where you are, but in the US, wires are usually connected using electrical nuts. Tug gently on each wire going into the nut -- there shouldn't be any give. If there is, remove the nut, arrange the wires so that they're parallel, use a pliers to twist them together, then screw the nut back on. People often wrap electrical tape around the open end of the nut, but it isn't required.

This looks to be wired correctly prior to the current code 2014 with the exception the white wire in the second box should be marked with tape black or red just not green or white. If there are metal boxes and the boxes are grounded things would be fine but you probably have plastic boxes so adding the ground to the switch would be a good idea. You did not mention if there is a ground in the second box there should be. And it should be connected to the switch also. The main difference in your wiring and current code is that the neutral needs to be run to the 2nd box with the ground and other 3. With the grounds added to the switch(s) it would be safe, with the whit leg identified as a Curren carying conductor, this would be up to code prior to 2014. your state may be on an older version of code so depending on that it could still be up to date with the changes.