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.
Drawing 5.8 uA for a few days would probably be OK especially if you set the cut-off a little higher, but the problem you're likely to have with that regulator is that the quiescent current is about 30uA while shut down so it'll be higher while running, possibly in the order of milliamps (it doesn't seem to be specified that I can see).
It sounds like you may not have considered that while the AVR on the Arduino in sleep mode may only be drawing 5.8 uA the regulator will continue to draw it's usual quiescent current. There's two things I'd recommend:
For something robust there are external under-voltage protection circuits you could use. For example I found a Linear Technology application note on a LT1495 based protection circuit that draws 4.5uA.
If it's for your own / general hobby use make the setup as described and set the AVR to go to sleep at 3.1 V and make some measurements to see how fast it's dropping towards 3 V. Then you can determine a good cut-off point to use that gives you an acceptable amount of time to remove the battery before it's damaged.
They'll be a bit of compromise between the cut-off voltage and how long it can survive without being recharged, but most lithium batteries rapidly decline below 3.2 V so you probably won't be missing out on a lot of charge.
Best Answer
Modern Lithuim-ion Polymer batteries don't like being discharged to 2.7V, and there's little to gain from going below 3.5V anyway because at that voltage there is almost no capacity left.