How do inductor specifications impact their effectiveness in a boost converter circuit?
If the datasheet recommends a 1uH inductor, can I substitute a 2uH inductor?

You can but you'd get a lower ripple current or the converter might not work as well as it would with a 1 uH inductor.

How do I know how much ripple current is acceptable?

That depends on your load and output capacitor. The ripple current causes a voltage ripple (through the output capacitor's series resistance, ESR) on the output voltage. That will be the supply ripple for the circuit operating on that voltage. Some circuits are very sensitive to supply ripple, others ignore it completely.

.. pull 3A from the power supply, can I use any inductor,.. maximum DC current greater than 3A

No, you need an inductor which can handle more than the expected peak current through the inductor. The 3 A delivered by the converter is the average current. The current through the inductor could be a lot higher. Especially when the a small input voltage needs to be converterted into a high output voltage. Then the inductor's charging cycle is long but the discharge cycle is very short.

but how do I know how high the DC Resistance can be before it negatively impacts the converter circuit or battery life?

You do system calculations and then you know.

Self Resonant Frequency

You don't "pick" a SRF, you just "don't pick" inductors that have a SRF which are too close to your switching frequency.

As commented, this is quite broad. Typically as a hobbyist you don't need to worry too much about all this. In general you would just buy a boost converter module that fits your needs or use an IC that can do what you need. Then just follow the chip manufacturer's datasheet. If you do what is suggested in the datasheet you will also get the performance as listed in the datasheet.