Can you tell me exactly what happens when you set the oven to bake? I understand that the oven is not turning on, but what do you see on the control board at that point? Both F2 and F3 stand for a problem with the oven sensor, however they contradict each other - one means the sensor is closed, the other means its open. Anyway, let me know about the board please.
As far as the oven igniter...
There are two oven igniters available - round and flat. One of them will work for you, you'll just need to take a look at the old one and see which one it is, no need for the model number.
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Round oven igniter
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Flat oven igniter
If the broiler is located under the oven, then both bake and broil cycles share the same igniter and burner. If the broil is inside the oven up top, then it has its own igniter and a separate burner.
There is no way to manually light the oven. Here is how it works (simplified version). Your control (whether an analog thermostat or a digital board) sends the signal to the igniter, the igniter begins to glow. When the igniter gets hot enough (30-90 seconds) it passes a specific amount of amperage to the oven safety valve. The valve then opens and lets the gas in to the burner. Igniter then ignites the gas. When your oven gets to your set temperature, the control shuts down the igniter, which in turn kills the valve. Then when the oven temp drops 50 or so degrees below set temperature, the control senses it and tells the igniter to start up again, etc. It cycles on and off this way until the bake cycle is turned off. So, where were we... Oh yes, if the igniter is not on and producing enough amperage, the valve will remain closed. That is why there is no way to turn the oven on manually.
Igniters break, but more often they get too weak. They would still glow, but won't pass enough amperage to the safety valve. If the igniter glow is not bright white, it's a good bet the igniter is gone and needs to be replaced.