You can get a Nitecore D2 or D4 to charge the gumstick: https://smile.amazon.com/gp/product/B00KW2ZDJO/ or https://smile.amazon.com/Nitecore-9004666-Industrial-Charger-Universal/dp/B00KGKEQ1S/

Ah -- an external charger is an easy implementation. You buy just one LiFEPO4 charging module, and a holder for your cell. When you need to recharge a cell that is in one of your devices, you just pop it out and put it in your external charger. This was you don't have to buy and integrate a bunch of chargers.

You can also buy and use one of these, which additionally monitors battery health (I have one):

https://www.amazon.com/Nitecore-Charger-Universal-Rechargeable-Batteries/dp/B00KW2ZDJO

If not implementing an external charger, your general topology is as follows:

1. Your battery is connected to the B+ and B- pads of your charge controller.
2. The out+ and out- of the charge controller are directly connected to VCC / GND of your switch mode power supply.
3. The output / GND of your power supply go directly to your load (the device power inputs). You turn the device on and off by connecting/disconnecting the enable pin of the power supply to the out+ from the charge controller.
4. To charge, you apply a voltage to the IN+ / GND on the charge controller (it has a total of 6 pads usually!). Typical voltage range accepted is 4.5-6.5V, some will take a little higher (useful for solar charging). The connector that delivers that voltage is more or less irrelevant, so you can feel free to use whatever combination of plug types you like.

For cell architecture, I'll give you a summary of the whole story first, then conclude with a recommendation.

Using Li-ion cells in parallel is usually a tradeoff between cost (it may be cheaper in some contexts) and reduced cell lifetime (lithium cells in parallel have some minor issues). Different cells might be charged to marginally different voltages, or discharge at different rates due to random manufacturing variance -- this can cause the cells to charge/discharge into each other when connected in parallel (not ideal). Also, if connected to a single charger, it has no way to detect the small differences between cells while charging. Moreover, for the battery (a battery is a combination of cells) to fail, only 1-of-N of the cells need to fail. So you're typically better off to use a single large lithium cell than multiple smaller ones in parallel. That being said, some commercial devices do this anyway -- I've seen this architecture in external cellphone batteries especially. I would expect a regularly used array of 2 cells in parallel to last something like 1-2 years before failure, which is OK, but this gets worse as you put more cells in parallel. A single larger cell is usually (but not always) cheaper and should last around 3 years.

A better architecture is to use 2 cells in series, as they cannot charge/discharge into each other. The battery still fails if 1-of-N cells fails, though. Also, the charge circuitry becomes more complex, but can manage each cell charge individually to maximize cell lifetime before failure. These battery charge controllers are labelled by the number of cells in series and parallel, for example 2S1P is two cells in series, with no extras in parallel. Anyway, the most common arrangements for hobby electronics are 1S1P (just a cell), 2S1P (~7.6v battery) and 3S1P (pretty close to 12V so quite convenient). I've seen up to 8S1P for e-bike batteries. Anyway, you may want to avoid the complexity here by just using a single cell and converting the voltage -- mistakes in wiring tend to destroy charge controllers.

That's most of the story. Now for your case, the optimal thing to do is probably to use a single cell (the largest you can fit), and use a buck-boost converter to drop the voltage down to 3V. The one I linked can operate with input voltages from 2.5V to 15V, and outputs either 3.3V (generally close enough to 3V) or 5V depending on the version you buy (the difference is probably just a resistor value somewhere on the boards). However if you do need more battery lifetime than one cell can provide, and wish to use 14500 cells, I would suggest using 2 cells in series (7.4v to 3.3v via the converter), and physically removing them from what you built, to charge them in an external charger like the Nitecore I linked. It's pretty hard for anything to go wrong this way, and the cells become standardized replaceable parts in your application. It might not be as 'cool' as a custom charging circuit, but it's very practical, and you can even predict cell failure based on increasing internal resistance, as displayed on the charger!

Better to ask questions here, because then other people can search for it. I'll answer as time permits -- when there's a slow period at work I answer questions here. It's good practice communicating engineering decisions (competing optimums, and so on) to people that have different backgrounds. I have to do this with management and executives as part of my job -- stuff like whether to spend the extra $$$ now to develop a good algorithm that grows in resource consumption with the logarithm of the number of users, instead of (for example) the square of the number of users. Usually management just wants to throw more resources at the problem, but I want to 'build it properly'. Working out what's actually best for the company is a horrible multidisciplinary problem with a lot of ego that needs to be put aside :D

It should, but I've seen it recommended to got for something like a pNitcore D2](https://www.amazon.com/Nitecore-Charger-Universal-Rechargeable-Batteries/dp/B00KW2ZDJO/)

Are you using the original Sony gumstick? Those are almost certainly dead by now.

You might be able to partially revive them using a smart charger, like the Nitecore D2, but they won't last nearly as long as expected.

You can get replacements fairly cheap now, too!