Electronic – Powering circuit directly from USB while charging li-ion battery side by side

LiIon is usually charged at constant current until a max allowed voltage is reached and then is held at that voltage while current tails off under "control" of chemistry of battery until Ichg = k% of Imax where k% is chosen according to longevity or max energy concerns.50% or 25% of Imax gives longer life. 10% or 5% tail gives max capacity but lower life.

Lowering Vpedesatl by 0.1V greatly assists battery life.

Discharging to higher cutoff voltage aids cycle life.

LiIon also has calendar life and starts self destructing from day one so a lightly used battery still dies.

Best cycle life is achieved by stopping charge when Vpedestal is reached and systen changes from CC to Cv. By monitoring voltage this point can be observed. You could even do a "dumb" system that simply watched delta Vbat and declared constant V when delta fell to zero. Only slightly more than a comparator and an RC delay in one input would achieve that.(While Vin is ramping a delayed vin is lower. When Vin pedestals the delayed Vin almost catches up. An offset voltage is needed to allow comparator towork).

LiIon cells mechanically flex the cell as metallic Lithium is "plated" in and out of the cell*. Cycle life is in large part due to battery beating itself to death mechanically.(This is why LiFePO4 lasts much longer and has lower capacity - the material is held in an Olivine matrix that maintains constant shape as active material is moved in/out BUT it takes up some space. )

*Note: Bill Dubuque has suggested that this sentence would be better replaced with " 'LiIon cells mechanically flex the cell as Lithium ions are intercalated".
The distinction is a finer one than may be apparent. However, it is true to say that if you cut open a LiIon cell you would not usually find metallic Lithium in it. Bill notes that this makes primary Lithium cells, which do contain metallic Lithium, a greater fire hazard than LiIon cells.
If you charge a LiIon cell with excessive voltage metallic Lithium will be 'plated out' and "vent with flames" mode usually occurs at about the same time.

Charge to CV level as often and as soon as possible.

If charging all the way their "disconnect message" is a sign of bad ethos. They are probably trying to minimise the risk of fire without telling you.

For longest storage life (as opposed to long life in regular use) storing at a lower voltage than Vmax is in order. Probably at about 3.6 V and only about 30% state of charge. The various Mars Rovers use LiIon batteries and have a design life of about 8000 cycles - but charge to about 3.6 - 3.7 V maximum.
8000 / 365(~=) ~= 22 Terran years.

There are many charger ICs out there. MCP73831 is one of the cheapest ones.

I have learned a lot from the site below over the years (and I'm involved in making charging, discharging, emulation circuits for Li-polymer batteries).

http://batteryuniversity.com/learn/article/charging_lithium_ion_batteries

The typical parameters for many lithium-ion and lithium-polymer cells are below. This is just typical specs. You need to read the datasheet for your cell.

1. Use a constant voltage source at the specified voltage (e.g. 4.2V or 4.35V) with built-in current limiting set no higher than the manufacturere's specified max charge current
2. Output voltage must have 1% absolute accuracy
3. Charge at a max rate of 1C (C = Ah capacity of battery). Example: A 100 mAh battery can be charged at 100 mA.
4. Stop charging (full charge is reached) when charge current falls below 1/10C. Example: A 100 mAh 4.2V battery is fully charged when you apply 4.2V to it and the current is less than 10 mA.

Again, these are just typical numbers.

There are critical safety issues that you must also address:

5. Do not allow the battery to below some minimum voltage, often 2.5V. If battery ever goes below minimum voltage, throw it away because it is damaged. Do not charge it afterwards as it can overheat, puff up due to high internal pressure, and there is a high risk of rupture and fire. (Personal experience.)

6. Do not exceed the manufacturers specified discharge current or charging current. The battery can overheat and go into thermal runaway.

7. Do not charge it higher than specified voltage + 1%. You can plate lithium metal onto the electrodes and cause a dangerous condition.

Commercial batteries have a safety IC controlling MOSFET switches that disconnect the cell to try to prevent conditions 5, 6, 7. Hobby batteries do not have such safety circuits.

Also, battery compartment design is important. You need to:

A. Leave enough room for the battery to swell slightly. Over its life it will "puff" slightly. A proper datasheet will tell you how much to expect. Cheaper cells often puff more over time.

B. May need some cooling if you are running at max discharge or charge rate (or exceeding it!).

YOU MUST READ THE DATASHEET TO MAKE SURE YOU MEET THE MANUFACTURERS REQUIREMENTS

Read that link and other references and absolutely read the manufacturers datasheet and you will gain plenty of knowledge. Hope that helps, -Vince