That diagram looks right to me appart from the power supply only being 1V, in your circuit it will be 12.
The diode can be any rectify diode 1N400x where x is any number 0 to 7,
The FET needs to be a logic level FET there are many of these. For example a 2N7000
Yes fine. Having a closer look at the data sheet the exact type of FE from that family is the NDS7002A.
There are many other logic level FETs like IRL520N
2n7000 or nds7002 is not a good choice of mosfet for switching 0.25A from an arduino. The absolute max drain current is only 200ma for the 2n7000 and 280ma for the nds7002a. Ths datasheet only quotes Rds(on) for 75ma drain current at Vgs=4.5V which implies that the mosfet drain current may saturate well before 0.25A unless you apply a higer gate voltage. I would use a bjt such as BC337 for that application. Alternatively, if making a pcb and using smd components, a mosfet such as AO3400.