Can I use multiple voltage regulators in a circuit safely?
No, this is not a good idea.
Check the datasheet of the regulators, but most don't like reverse voltage.
Parallel voltage regulators can't be counted on to share current. One will always have a little higher setpoint than the other. This could also lead to instability, depending on the nature of the controllers in the regulators.
A better way to solve this is to combine the two voltage sources before a single regulator. At this low voltage, you can use Schottky diodes. Put one diode in series with each voltage source. Power will then automatically be taken from the higher of the two voltage sources. Make sure to put something like a ceramic cap after the diodes physically close to the input of the regulator.
You can still use multiple regulators to spread the dissipation and to reduce voltage drops to distant parts of the circuit. You bus around the higher voltage out of the diodes, then regulate that locally as needed. However, you don't tie the outputs of multiple regulators together. You have each power a different part of the circuit instead.
If you want to minimize local dissipation, you use a buck switcher after the diodes. This makes a little more than the minimum input voltage of the regulators. You filter that a little and bus that around. Then you make the regulated voltage from that as needed locally. For example, if using a 5 V LDO that requires 5.5 V in, you might bus around 6 V. Each local regulator would then be 83% efficient.
Analog Devices has a great article that discusses current sharing in parallel voltage regulators here. The gist is that you need to add extra circuitry in order to have the regulators share current relatively equally. A quote from the article is shown below. They are discussing Low Dropout Regulators (LDOs), but the same applies standard regulators like the 7805.
I would also worry even more about current sharing when powered by two different voltage sources, since the regulators would be even less matched in that case. Also, in the case of one of the voltages being disabled, if the regulator does not have reverse current protection, current can flow from the output back to the voltage input that is off.
Current sharing with linear regulators is traditionally not as simple as connecting the parts in parallel. Two voltage reference-based linear regulators set to the same output voltage and with the outputs tied together will not share current equally. An LDO's output voltage is determined by the reference voltage multiplied by a gain factor based on the feedback resistors. Due to tolerance errors in the voltage reference and feedback resistors, the output voltages will be mismatched. With unmatched outputs, the LDOs will not share current; one LDO will provide the majority of the current until it hits current limit, thermal limiting or its output droops low enough for the other LDO to begin supplementing its current. These three situations present circuit operation challenges and can pose reliability concerns, leading to possible premature failure of the overstressed LDO.
This will depend on what the datasheet of the regulator says.
In general: No, you cannot operate them in parallel, because supply and load changes will have different effects on your regulators, and thus they will start regulating "against each other".
Even more basic: No two regulators are exactly the same, even under perfectly constant conditions, so one is bound to constantly "pull up" the voltage, while one is constantly pulling it down.
Whether you can just put them in parallel with one unpowered: In general, I'd say "no". Again, if the datasheet doesn't contain clear information that this is possible, it probably isn't. Read the datasheet.
LM7805 (and clones thereof) are relatively "dumb" and slow, so you might be in luck and operating them in parallel might under non-oscillating load conditions actually work relatively well. Do not power one but not the other: current will run through the unpowered one!!
So, no, don't do that, especially not with 7805s. Instead, consider just having one regulator, whose input you switch between the different sources.
Also, it's probably a very bad idea to regulate 16.8 V down to 5 V with a linear regulator: Your waste heat is (16.8 - 5) V · I_out = 12 V · I_out, and if your output current is let's say 250 mA, you're already burning 3 W, which heats up your device by 36 °C. If you use more than 250 mA, you get hotter.