Electrical – What would happen if I connect a battery which has lower voltage than output of battery charger

A1: The block diagram is a good start. A schematic would be even better.

A2: 5V is fine. The data sheet shows it is 97% efficient with a 5V input (see figure 7 in http://www.ti.com/lit/ds/slus534e/slus534e.pdf)

When charging lithium batteries in series, you need what's known as a balance charger to keep the cells at similar voltage. One way that some chargers do this is by stopping the charge of both when one is full, discharging the full one until they are even, and then charge them both up again. There are fully featured chargers meant for the remote control hobby, but they require setting the charge options on the screen before every charge. I cannot seem to find any "plug and play" balance charger boards similar to the TP4056. I am fairly sure that ICs exist that will handle the buck/boost power supply and the balancing, but I can't seem to find one and you would need to manufacture your own circuit board, and the chip is almost certainly surface mount.

That being said, there are a few workarounds. One is to have the batteries in series for operation and parallel for charging, and use either one or two single cell chargers. This can be accomplished through a 3PDT switch, and a good explanation is here. Note that you don't need to use the same charger, your TP4056 should be able to be substituted.

A simpler and more elegant way to do it would be to use a boost mode power supply, and don't put the cells in series. You can attach multiple of your current cells in parallel, so that if you have two cells and are drawing 5W, it would be 2.5W per cell or ~0.67A. There are many chargers that are designed for this application, handling both the CC-CV charging and the boost regulator for the output, such as the Lipo Rider Pro from Seeed Studio which will do 1A output. This is what I would recommend.

Alternatively, you can buy more powerful batteries that will put out more than 1A instead of putting multiple in parallel.

Or if you want to take all the fun out of this project and just buy something that sort of works, there are USB battery banks available from China that will take care of all the charging and discharging of 18650 LiPo batteries. I have seen banks that will take from one to six 18650 cells in parallel, and some of the larger ones can do as much as 3A output. Here are a few examples, and a video review. Their build qualities and user interfaces look pretty bad, but apparently they work.