Short answer: No, because your "5V USB Charger" is not a charger – it's a power supply. No, you can not. You must read this: Safety Concerns with Li-ion. (There is a reason why R/C enthusiasts put Li batteries into flameproof bags, when they charge them.)
Probably not directly. A 5V charger has a specified voltage. If 5V are applied to a discharged Lithium-Polymer battery (around 3V), the current going into the battery might be several times over the maxium tolerable charging current for a 380mAh battery, which would be 380mA (better is 100mA).
In order for current to flow there needs to be a useful voltage difference. There is a voltage difference between 5 and 4.2 and 3.7. If you charged with 3.7 there would be no voltage difference, no current and no charging. Less than 4.2 volts and it won't charge fully.
Charging a 3.7v battery with a 5v charger. Here's everything about using a 5 V charger to charge a 3.7 V cell phone battery being safe to use: It's not safe to charge a 3.7 V cell phone battery with a 5 V charger for both the equipment and personal safety.
Secondly, a lower voltage charger will never be able to produce enough power to fully charge your 6V battery. It might charge your battery enough for you to get your device jump started, but that's about it.
Almost ALL THE TIME mixing voltages like you did is a bad idea. It could result in immediate failure that could result in smoke or even fire.
It could result in immediate failure that could result in smoke or even fire. Other times, it results in a shortened life of the appliance. Very rarely you will find that this works, but only if the difference is small, and usually only on simple appliances or products.
No, a 5v adapter is never recommended to power a 6v device. Because we need a voltage higher than that of the battery before it will charge properly. As a device is required to run from the voltage rating it is given, hence if we use a 5v adapter for a 6v rated device, the device won't start at all.
A 3.3 V device can interface with a 5V device without any additional components. For example, a logic 1 (HIGH) from a 3.3 V device will be at least 2.4 V. This will still be interpreted as a logic 1 (HIGH) to a 5V system because it is above the VIH of 2 V.
Most of the times it won't. Cannot tell for sure without checking the internal hardware of the repeater. But if it works without generating heat it will be fine as far as I know. The 5V adapter might generate more heat than the 6V as sometimes the lower voltage will increase the current draw.
6V batteries need to stay below 7.1V to avoid gassing, and typical charge voltages are 6.9V (float) to 7.5V (bulk charge).
The easiest way to convert a 5V analog signal to a 3.3V analog signal is to use a resistor divider with an R1:R2 ratio of 1.7:3.3.
It could result in immediate failure that could result in smoke or even fire.
You need to have that order of "leeway" voltage across the resistor in order to allow for variations in the LED voltage, so for an LED with a conduction voltage of 3 V, it is generally not practical to operate it from less than 5 V.
It depends on the specific battery and charger in question. If the 5V charger is specifically designed for use with a 6V battery, then it should work just fine. However, if the charger is not specifically designed for use with a 6V battery, then it's possible that it could damage the battery or cause other problems.