Does Low Power Mode Affect FaceTime and WhatsApp Connectivity?

As you noted from the system profiler, your battery is close to getting ready to fail.

Battery is in the "Replace Soon condition"

Your best guess is accurate in that the "battery runs out before OS X has a chance to finish saving the full memory image to disk" This memory image is used to enable Safe Sleep. Without it, applications start from scratch.

A thing to try before replacing the battery is to attempt to calibrate the battery.

Calibrating MacBook Battery

PowerBook G4 (15-inch Double-Layer SD), MacBook (all models), and MacBook Pro (all models)

The battery calibration for the PowerBook G4 (15-inch Double-Layer SD) and any model of MacBook or MacBook Pro has been updated because of a new battery released with this computer. With these computers, follow these steps to calibrate your battery:

1. Plug in the power adapter and fully charge your PowerBook's battery until the light ring or LED on the power adapter plug changes to green and the onscreen meter in the menu bar indicates that the battery is fully charged.

2. Allow the battery to rest in the fully charged state for at least two hours. You may use your computer during this time as long as the adapter is plugged in.

3. Disconnect the power adapter while the computer still on and start running the computer off battery power. You may use your computer during this time. When your battery gets low, the low battery warning dialog appears on the screen.

4. At this point, save your work. Continue to use your computer; when the battery gets very low, the computer will automatically go to sleep.

5. Turn off the computer or allow it to sleep for five hours or more. Connect the power adapter and leave it connected until the battery is fully charged again.

Tip: When the battery reaches "empty", the computer is forced into sleep mode. The battery actually keeps back a reserve beyond "empty", to maintain the computer in sleep for a period of time. Once the battery is truly exhausted, the computer is forced to shut down. At this point, with the safe sleep function introduced in the PowerBook G4 (15-inch Double-Layer SD) computers, the computer's memory contents have been saved to the hard drive. When power is restored, the computer returns itself to its pre-sleep state using the safe sleep image on the hard drive.

If your computer continues powering off completely in-between steps 4 and 5 after a couple of trys, replacing your battery may be the only way to reliably get the safe sleep functionality back, other wise you may need to manually configure your MacBook to go into Safe Sleep when you put it to sleep manually well before the hard power-off occurs. A Utility such as SmartSleep might help greatly.

Update: this was added starting in iOS 11 (Apple Video).

In brief:

1. Open Settings
2. Select Control Center
3. Tap Customize Controls
4. Find Low Power Mode option and tap the green +

more info

Original:
I was very surprised to open Control Center on iOS 9 and not see a battery icon! Apple manages to fill other things in very nicely and that would have been very logical (from my view).

Unfortunately, as of iOS 9.0, there is no simplified way to toggle Low Power Mode. As you discovered the only way to enable it is to open the Settings.app and then opening Batteries. If you'd find it faster, you can now search the Settings app by pulling down past the "top" of the list in the app.

In addition, when your phone hits 20% battery, it will alert you and offer to enable the feature.