1) This should be fine with a low-power solar panel, but it will charge very slowly.
First a bit of theory. Charging LiPo batteries needs to follow two rules:
- the end voltage (called float voltage) must be lower than or equal to 4.2 V
- the charge current must never be higher than a certain value (typically 1C, meaning 100 mA for a 100 mAh battery, 1 A for a 1000 mAh battery, etc.)
If you have a 4.2 V regulator then the battery voltage can't go above 4.2 V. The charge current will decrease to zero as the battery voltage approaches the regulator target voltage.
When using a wall adapter, you need something to limit the charge current, even if the adapter has a low power rating. Otherwise you might damage both the battery and the power adapter. This is because a LiPo battery has a very low resistance and there will be a rush of current when you connect the power.
But with a low-power solar panel you could rely on the fact that your particular panel is incapable of supplying more than 1C of current to the load. Then your solution might work but it will be very, very slow to charge. That's because whenever you try to draw too much power from a solar panel, its voltage collapses abruptly and you get no power at all from it.
For a 5 V, 1 W solar panel in good light, this is what happens to the voltage and power when you increase the current draw by decreasing the resistance of the load:
- 20 mA: 6.0 V (0.12 W)
- 100 mA: 5.5 V (0.55 W)
- 200 mA: 5.0 V (1 W)
- 250 mA: 0.1 V (0.025 W !!)
So you would have to get an oversized panel and use a resistor in series with the battery after the voltage regulator to limit the charging current. This will work but it is wasteful.
For good efficiency you need a circuit that reduces the charge current when the solar panel gets less light. See for instance this one (tutorial). What this board does is reduce the charge current whenever the panel voltage goes below 4.75 V, and increase it when it goes above. It also handles the charge current limit for the battery. For further reading google "Maximum Power Point Tracking".
2) You only need to interrupt one wire to open a circuit. One diode is enough. You don't need two switches for your lights.
3) This would work with the reserves outlined above. But there is another danger: LiPos should NOT be discharged too deep, otherwise they will be damaged. So you really should use a protection circuit to cut power to the load if the battery voltage is too low, and also to make sure that the load current isn't too high. Solar charger boards like the one I mentioned should do all that.
I recommend reading the tutorial above, and especially the "design notes" section if you want to understand more about solar panels.
A regulator is a good, simple idea; especially if you already have a regulated voltage (buck converter or similar) present. If that is the case, you can use this as the threshold voltage and simply scale down the battery voltage accordingly.
If not, you need to build a small circuit that produces a voltage independent of the battery voltage: The easiest solution is to utilize a zener diode in reverse bias (and a current limiting resistor.)
simulate this circuit – Schematic created using CircuitLab
Make sure to calculate the values for your specific design. There are also different types of comparator outputs, perhaps you need to drive the LED differently.
You can determine the value of the voltage divider by considering that the voltage at the negative input needs to be the same as the threshold voltage (at the positive input). If I understood you correctly, you want the LED to turn on for battery voltages lower than 3.3V (V_th); that means you calculate
$$ \frac{V_{threshold} * R3}{R2 + R3} = V_{@inverting} $$ where you arbitrarily choose R3 to have some resonable resistance (based on how much current it would draw) and your arbitrarily chosen \$ V_{th} \$ of 3.3V.
Consider using two comparators so you can have two thresholds; if you use single package with two comparators this hardly takes up more space.
Best Answer
The device, or circuit, you are looking for is called an inverter. You are right that it has a transformer connected backwards, but it also has a lot more besides. A block diagram of a typical inverter system might look something like this:
simulate this circuit – Schematic created using CircuitLab
The H-Bridge is used by the controller system to generate an alternating current (AC) waveform. The transformer then steps the waveform up to the target voltage.
It's not a nice clean sine wave like you'd get from the mains, but instead what is called a "Modified Sine" wave (green line in this graph):
Due to limitations in the way the H-bridge works there has to be a gap (or dead zone) between driving in one direction and driving in the other. This gives the waveform its characteristic three-level appearance.
Inverters are not a very efficient system, since a proportion of the power is wasted as heat and even noise (they usually hum). It is far better to find a low voltage solution to your problem. Most appliances nowadays (excluding normal white goods) actually run on low voltages, and contain a power supply to reduce the voltage down to the typical 12V/5V/3.3V that you find everywhere now. A much better solution is to find products that allow you to bypass that power supply and provide the needed voltages (or a subset of them - typically 12V is used) directly to the device.