Saturating a N-Channel MOSFET
The gate threshold voltage of a FET isn't all that useful in a design.  What you care about is how on the FET becomes when your gate voltage is applied.  This is the Rdson spec, which may be listed at multiple gate voltages.

You are using a 2N7000, which is not a particularly impressive FET.  At 4.5 V gate voltage, it gets down to 5.3 &Omega;.  That may be OK if the relay coil is significantly more than that, which it probably is.  Let's say the relay coil is 80 &Omega;.  In that case the total resistance will be 85.3 &Omega;, the current at 5 V will be 59 mA, the drop accross the FET 311 mV, leaving 4.7 V accross the relay.  Not great, but should be good enough.  However, note that this was a example.  Do the math with whatever resistance your relay coil really has.

To drive a FET like this, there is no need for a resistor between the digital output and the FET gate.  Just connect them directly.  The flyback diode needs to be rated for whatever maximum current will go thru the relay coil.  A little less will be OK since the diode will only be conducting the current for a short time.  Unless you have a "large" relay (in which case the 2N7000 probably won't be good enough), the 1N4148 you show should be good enough to catch the kickback current.

A few problems with your circuit:

You need to rectify the 12 V out of the transformer before running it into the 7805 regulator.  Four diodes in a full wave bridge configuration would do nicely.  Use something like 1N4001, or a integrated full wave bridge module.  Put a cap after the bridge to hold the DC level high enough for the regulator between the line cycles.

The peak voltage of a 12 V sine will be 17 V.  Minus about 1.4 V drop for the full wave bridge, you are left with 15.6 V at the peaks.  When unloaded, it could be a few volts more.  This is well within the voltage spec of a 7805, but will cause it to dissipate a lot of power.  If you can find a "6V" transformer, that would be a better choice.

The relay can most likely work from AC, but the FET can't switch AC.  The top of the relay must also go to the output of the full wave bridge instead of the transformer directly.  Keep in mind that the relay will then see the higher unregulated voltage, which could be 15-20 V with this transformer.  That might be OK for a "12v" relay, but you might need to put a resistor in series with it to stay within the maximum allowed coil current spec.