Electronic – How much energy can this battery store

There are a number of ICs that will do these functions. I would go to the Maxim and Linear Technology sites. The downside is that most of these devices is that they are only in surface mount packages and a lot of them come in packages without leads (QFN, LGA).

A part like the LTC4088 is nice. It has an input for the battery, an input for the USB (or wall adapter) and a switching regulator.

You could also use separate ICs (but you are probably better off with a single). There is a device that Linear Tech calls a PowerPath controller (LTC4412) that performs the switchover function. You could add a MAX1555 Li-ion charger for a low cost charger and add the LTC4412. Also there are devices called "Ideal Diodes" that will perform switchover. These devices are FETs so you get a much lower voltage drop than a Schottky.

The best solution depends on a couple of things you haven't mentioned -- system current, battery capacity and recharge rates.

The discharge rate dependes of several factors and will vary from manufacturer to manufacturer.

A good Li-Poly typically has 2-5% self-discharge rate per month. If it has a PCM (protection circuit module), that module will draw some current, expect less than 1uA for such a tiny cell. Parts of lesser quality may draw 30uA or more.

Self-discharge increases with temperature. I believe it also varies with cell construction (cylinder, prismatic etc.) Quality varies a lot from manufacturer to manufacturer so you have to ask for the exact specs.

UPDATE

I don't have much experience working with the current levels you mention. Wouldn't the ground current in the charger consume most of the 10-20uA? There is a lot of interest in energy harvesting and someone else can probably give a better answer. Slow-charging the battery is OK provided you cut off at 4.2V. My guess is that 10-20uA is not enough to offset self-discharge and other losses, but I'm frequently wrong.