SMPS Current Sense Resistor Selection
I believe I understand where your confusion is from. The device limits its max inductor current to 1.8A. Then why on the world did the reference design protect it from sourcing out 5A? 

I will answer your question and clear the confusions in 3 sections:


why did the reference design protect it against 5A while the device's max inductor current is limited to 1.8A?


If you read the schematic and app note carefully, you will notice that the 5A and the 1.8A are NOT the same current. 
The 5A is the battery's max current. i.e. the load current: current flow from Cout1 to the load. (e.g. the battery you want to charge)
The 1.8A is the inductor current limit. i.e. the upper PMOS current limit. 

Is it possible the load current is higher than the inductor current? Yes, if the circuit is in transient. Consider the example here:

At time t0, there is no load, the SMPS is in steady state, regulated at 5V. average inductor current =~ 0A. Load current = 0A.

At time t1, a cap with 0V in series with a 0.5ohm resistor is connected to the SMPS' output cap, Cout1. inductor current can't change immediately, therefore remains small. Load current = (5V - 0V)/0.5ohm = 10A.

If this 10A is sensed by the current sensing resistor Rs1, the SMPS's lower FET will stop at once, and not switching again until this current is less than 5A. 

Once the load current drops to lower than 5A, the regulator will switch again. At that time, the 1.8A limit will kick in and essentially making the voltage loop to a current loop until the transient is over.


"I'd like to limit a 350mAh battery to a 1C charge rate, but I'm not sure if I should trust the value of 0.05 / 0.35 = 0.143Ω that I get"


If you want to limit your max charging current to 350mA, then yes, use 143mohm, though I wonder that's a standard resistor value. You will most likely use something close, e.g. 150mohm. On the other hand, what max current should you use depends on your battery. Hopefully you can find it on your battery's documentations.


"if I'm planning to use a small solar cell which is unlikely to ever produce 350mA @ ~4.1V, can I replace this sense resistor with a 0Ω jumper?"


Maybe yes, but I don't recommend. 

There are a few reasons:


As stated before, high load current may happen during transient, and your current sensing resistor protects your battery.
A protection is for unexpected events. Even during normal operation, if something went wrong (e.g. some one plugged a power supply to your boost's input), at least there is a protect there. 
Without the sensing resistor, you may end up with larger Cout by parallel the input cap of the load directly to Cout. This changes the transfer function of the boost converter, and may or may not cause issues depends on the loop design and control strategy. If a sensing resistor is there, a much closer pole zero pair is created and the impact to the loop is much less.


I hope that answers your questions.