Safe Lithium Battery Management
Batteries are tricky nonlinear devices. If you want to build a safe and reliable lithium battery charger, you need to know more about your batteries and your battery charger circuit. In general, lithium batteries are not interchangeable and not all chargers will work with all batteries.
There are also significant differences between lithium (probably not what you mean), lithium ion (probably what you mean, sometimes written Li+ or Li-ion), and lithium polymer (sometimes written LiPo) batteries, and significant differences within the battery chemistries of these categories.
The datasheet I found for TP4056 does not say that this device includes undervoltage protection. It only provides charging. Perhaps other circuitry does on that evaluation board does. The appropriate discharge protection is a function of battery chemistry, and the threshold should set according to the battery manufacturer's datasheet (as a baseline, anyway).
A circuit that tests for undervoltage is probably measuring the battery under load, and will need to compensate for the battery's internal resistance \$R_{internal}\$ reducing the voltage at the terminals \$V_t\$. That is
$$ V_t = V_{oc} - IR_{internal} $$
So to answer your second question it's quite possible that \$V_t\$ = 2.5V is an appropriate cutoff for a battery with \$V_{oc(min)}\$ = 3.0V, if \$ IR_{internal} \$ ~ 0.5V.
TP4056 is a charger only, with no switched path from battery to output.
You can buy the same board on ebay here for $US1.19 and they do not mention an LV cutout: 5V Micro USB 1A 18650 Lithium Battery Charging Board Charger Module+Protection
TP4056 data sheet here
Charger out = battery = pin 5.
The battery +ve (B+), charger out (pin 5) and Vout (OUT+) terminals appear to be connected on the PCB.
The small IC may be a low voltage monitor and the larger one a low side MOSFET. If so, there are two resistors which would probably allow Vmin to be increased.
But the TP4056 is less flexible than slightly more costly ICs available locally. If you want a design to handle many battery sizes, a more up market IC may be better.