Do I have to deal with inrush current for my AC/DC converter?
When switching the power on, can I expect a high inrush current
destroying the transformer or the Full bridge rectifier?

There are two inrush currents associated with this type of power supply: -

Charging the reservoir capacitor
Transformer magnetization current overload causing significant inrush

The 2nd one doesn't affect the bridge rectifier because it only affects the transformer primary magnetization current so if you need to worry about your bridge surviving a power up event just rate it accordingly for supplying a high current for the first half cycle of applied AC into the reservoir capacitor.
As for the transformer inrush due to core saturation, if it's designed to run on the voltage you use then there shouldn't be a problem either providing it's a device from a reputable source. The inrush will disappear after a few cycles.

As for the transformer I have no clue. It seems it has a DC resistance
of 6 ohms. Is it safe to assume that for fast inrush currents, the
inductance of the transformer will have a high impedance and therefore
it will be not a problem?

No you can't assume the primary magnetization inductance will protect it because, it is this inductance that plummets during an unfavourable inrush situation; it causes the inrush. But, transformers are a rugged piece of well-proven tech and chances are very high that it will survive without a hitch (if it's a reputable component) for many years.
So, I don't think you need to worry about adding a series thermistor unless of course, your upstream fuse blows occasionally when the transformer primary is connected to AC.