Since flaps help lift the plane, would they deploy flaps if the plane was nosediving? And what would they do if the back pressure was too much? I know speed increases lift, but can full throttle make the plane difficult to level out?
No, flaps are never used during a nose dive.
We need to distinguish between two scenarios:
From the Boeing 737 NG FCTM (7.5 Maneuvers - Rapid Descent - Level Change (LVL CHG)):
Because of airspeed and altitude protection and reduced crew workload, use of the autopilot with LVL CHG mode is the recommended technique for rapid descents. Use of the V/S mode is not recommended.
Initiate a turn, if required, using HDG SEL. Set a lower altitude in the altitude window. Select LVL CHG, close the thrust levers and smoothly extend the speedbrakes. Autothrottles should be left engaged. The airplane pitches down smoothly while the thrust levers retard to idle. Adjust the speed as needed and ensure the altitude window is correctly set for the level off. During descent, the IAS/MACH speed window changes from MACH to IAS at approximately 300 KIAS. Manually reset to VMO as needed.
When approaching the target altitude, ensure the altitude is set in the MCP altitude select window. Altitude capture engages automatically. Adjusting the command speed to approximately LRC or 300 knots before level-off aids in smoothly transitioning to level flight. The pitch mode then controls altitude and the thrust levers increase to hold speed. Smoothly return the speedbrake lever to the down detent during the level off maneuver.
When descending with the autopilot engaged and the speedbrakes extended at speeds near VMO/MMO, the airspeed may momentarily increase to above VMO/MMO if the speedbrakes are retracted quickly. To avoid this condition, smoothly and slowly retract the speedbrakes to allow the autopilot sufficient time to adjust the pitch attitude to maintain the airspeed within limits.
When the speedbrakes are retracted during altitude capture near VMO/MMO, a momentary overspeed condition may also occur. This is because the autopilot captures the selected altitude smoothly by maintaining a fixed path while the thrust is at or near idle. To avoid this condition, it may be necessary to reduce the selected speed and/or descent rate before altitude capture or reduce the selected speed and delay speedbrake retraction until after level off is complete.
In case a higher descent rate or lower descent airspeed is required, the landing gear can be extended additionally to the speedbrake, but flaps are never mentioned:
Landing Gear Extended Descent
The rapid descent is normally made with the landing gear up. However, when structural integrity is in doubt and airspeed must be limited, extension of the landing gear may provide a more satisfactory rate of descent.
If the landing gear is to be used during the descent, comply with the landing gear placard speeds.
Accidental nose dive: this would fall under the general term of upset. The recommended procedure is described in the FCTM (7.26 Maneuvers - Upset Recovery):
Nose Low, Wings Level
In a situation where the airplane pitch attitude is unintentionally more than 10° nose low and going lower, the airspeed is increasing rapidly. A pilot would likely reduce thrust and extend the speedbrakes. Thrust reduction causes an additional nose-down pitching moment. Speedbrake extension causes a nose-up pitching moment, an increase in drag, and a decrease in lift for the same angle of attack. At airspeeds well above VMO/MMO, the ability to command a nose-up pitch rate with elevator may be reduced because of the extreme aerodynamic loads on the elevator.
Again, it is necessary to maneuver the airplane's flight path back toward the horizon. At moderate pitch attitudes, applying nose-up elevator, reducing thrust, and extending speedbrakes, if necessary, changes the pitch attitude to a desired range. At extremely low pitch attitudes and high airspeeds (well above VMO/MMO), nose-up elevator and nose-up trim may be required to establish a nose-up pitch rate.
Also here, flaps are not mentioned, so only speedbrakes should be used to increase drag (and increase the nose-up pitching moment).
Looking at the Performance Limitations of the Boeing 737 (page 3), we see that the Flap Limit Speeds are
+================+==============+ | Flap Position | Limit Speed | | | (KIAS) | +================+==============+ | 1 | 250 | +----------------+--------------+ | 2 | 250 | +----------------+--------------+ | 5 | 250 | +----------------+--------------+ | 10 | 210 | +----------------+--------------+ | 15 | 195 | +----------------+--------------+ | 25 | 170 | +----------------+--------------+ | 30 | 165 | +----------------+--------------+ | 40 | 156 | +----------------+--------------+
This behavior is the same in basically all aircraft as far as I know. With flaps down, the wings need lower loads than when "clean". So deploying flaps when in a dive can be catastrophic.
For an example of what happens when deploying flaps/slats at high speed, see the accident of Austral 2553.