Charge Li-Po battery with TP4056? (Or what else?)
Take a look at Microchip parts: MCP73841/2/3/4. I've been using a MCP73844 in comercial products for more than 3 years and they work great! Unfortunately they do not have protection circuit for cut-off, but you can implement one with some resistors, an op-amp, a shunt and a MOSFET.