The faults above 260C are probably occurring because you haven't increased the temperature limit. See https://duet3d.dozuki.com/Wiki/Gcode?revisionid=HEAD#Section_M143_Maximum_heater_temperature.
Dead time of 5.5sec is about normal, so I suggest you do not increase it. Dead time should only be increased either if tuning fails because it measure a dead time less than 1 second, or if the temperature oscillates when the heater is supposed to be maintaining temperature.
The "Temperature rising more slowly" message normally means that the A value in your M307 command is too high. This can happen if you tuned with the print cooling fan off, but you started heating with the fan on, and you don't have a silicone sock over the heater block.