230vac / 5vdc power supply design
Literally any diode can respond to a 50Hz wave. But try a full bridge rectifier that can handle >1.5A, eg from here.
You could, for protecting the circuits under test. But that demands other components and schematic. See the datasheet of the regulator and do a quick search on short-circuit protection, to see what fits you.
Some suggestions:


Bigger caps at the input and output of the regulator. Use electrolytic and MKT/MKP in parallel in order for the PSU to be able to respond both at low and high frequencies. 
At DC side, use a fuse >1A.
Search for a heatsink that is suitable. As you have already noticed, you have quite a lot losses. Input voltage of the regulator is \$V_{IN} = \sqrt{2} \cdot12= 17V\$. Maybe find a transformer with less output voltage?



Have you considered buy a phone charger for now and design a SMPS? you will gain much more knowledge and experience.