I know that older planes must be hand started but are newer GA planes even able to be hand started by spinning the prop? If so, how do you do it safely without cutting your hands off or being pulled in by the propeller?
In reality the engines in modern prop planes are not all that different (in some cases even identical) to the engines of the hand crank days. The reason many older planes were hand crank was simply to save the weight toll of a starter motor and battery to drive the system. Some older aircraft have no active electronics (the piper cub comes to mind). Remember the engines magnetos drive the spark for the ignition system unlike in a car where the spark is driven by the electronic system. As long as proper precaution is taken its more than possible. The FAA even explicitly discusses it in their airplane flying handbook (p 2-13)
A spinning propeller can be lethal should it strike someone. Historically, when aircraft lacked electrical systems, it was necessary to “hand prop” an aircraft for starting. Hand propping an aircraft is a hazardous procedure when done perfectly. The consequences of not mitigating the hazards associated with hand propping can lead to serious injury, fatalities, and runaway airplanes. All alternatives must be considered prior to hand propping an aircraft and, when a decision is made to do so, the procedure must be carried out only by competent persons who have been trained to accomplish the procedure, understand how to mitigate the hazards, and take all the necessary precautions.
They also provide a pretty clear checklist on how to do it.
The procedure and commands for hand propping are:
• Person out front says, “GAS ON, SWITCH OFF, THROTTLE CLOSED, BRAKES SET.” 2-14
• Pilot seat occupant, after making sure the fuel is ON, mixture is RICH, magneto switch is OFF, throttle is CLOSED, and brakes are SET, says, “GAS ON, SWITCH OFF, THROTTLE CLOSED, BRAKES SET.”
• Person out front, after pulling the propeller through to prime the engine says, “BRAKES AND CONTACT.”
• Pilot seat occupant checks the brakes SET and turns the magnetos switch ON, then says, “BRAKES AND CONTACT.”
As far as i know, you can not hand start a turbo prop (which some GA planes have) due to the higher RPM needed for start.
If so, how do you do it safely without cutting your hands off or being pulled in by the propeller?
The short answer is "very carefully."
A longer answer, and the way my father taught me to do it (but not necessarily how others might do it) when I was in my early teens was as follows:
Stand close enough to the propeller disk so that you're not leaning forward. The closer the better, but remember that if you're propping a taildragger the bottom of the disk is closer to you than the top. Beware if wearing clothing that has lot of loose fabric that might contact the propeller. The two times that my father was slightly injured was due to contact between the propeller and lose fabric when propping taildraggers.
Rotate the propeller with the switch off until the blade you'll be propping is at about the 10 o'clock position and just before a compression cycle, then switch on.
Place the palms of both hands on the blade about 85% of the way out from the center. The tips of your fingers should be at the trailing edge of the blade, but be careful not to wrap your fingers around that edge.
The next part was tricky but was necessary because both my father and I were small and light. It worked best for us if we swung our left leg (it may have been the right leg but I really can't remember since this was over 60 years ago) up into the propeller disk and then smoothly, briskly, and aggressively bring the propeller blade down while swinging the leg down and out of the way, continuing into a step away from the propeller.
Radial engines, because of their lower compression, were easier to prop than inline engines. However, I wasn't tall enough when young to handle their height.
The aircraft I hand propped the most as an adult was an 85 horse Continental on a J-3 floatplane. You started it on one mag, and that mag had a spark delay for starting that didn't fire until the bottom of compression rather than the top. You steadied yourself on the front of the right float with your left hand grasping the fuel tank filler cap and nozzle just in from of the windshield and propped with your right hand. Because you were propping from behind, you had to use your fingers over the blade trailing edge, but the spark delay obviated the danger of backfire.