Electronic – Input Bias Current in an OP-AMP and its value in a datasheet

1. Yes, the LM311's input bias current does flow out of the pins - the schematic on page 3 of the data sheet shows PNP transistors and this implies bias current flows from the pins.
2. Yes, the LM358 data sheet shows this current as a negative current but, the bias current will be flowing out from the pins just like the LM311.

Some op-amps, notably rail-to-rail input op-amps may specify this as "+/-" because they usually have NPN and PNP combined input stages. But if one of two (or maybe plenty) of data sheets don't obey the convention I don't think it's a big deal - if you have to deal with bias currents (on sensitive applications) you match the input resistances or use low enough resistances around the inputs to minimize the effect. What will remain is the input offset current which can always be presumed to be "+/-" because one input's bias current may be slightly less or more than the other input's bias current.

You are confusing a few things (as does your youtube). First, you need to distinguish between bias currents and offset current. The offset current is simply the difference between the two bias currents. If the offset current is significantly lower than the bias current, it says that the two bias currents are pretty much the same. If you look at the youtube, however, you'll see that the max bias current is listed as 100 pA, while the max offset current is 200 pA. In other words, there is no way to compensate since there is no way to tell exactly how the two bias currents will compare (the youtube guy got lucky).

In your case, though, you have an advantage. The independent bias currents which make offset current compensation difficult is characteristic of MOSFET op amps, while bipolar op amps do have correlated bias currents - and you are using a bipolar op amp. Please check the data sheet under Electrical Characteristics. For the TL972 the currents do correlate pretty well, so you can add a compensation resistor and expect some improvement. If you are going to do this, though, you need to calculate your parallel resistance correctly - in this case it is about 42k, not 140k. While you have done this and don't like the results you got, that does not change the fact that that's about as good as you're going to get. All this says is that the problem lies somewhere else, and fiddling with the compensation resistor will not help you understand it.

Once you've done that, though, it seems clear that you have missed the larger point of the youtube - offset voltage. Watch it again, and you will see him calculating the effect of Vos (input offset voltage) as Vos times the gain of the circuit. In your case, your gain is 10. And again, if you will read the data sheet, you will see that the max offset voltage is 6 mV. Multiply by 10 and you get a 60 mV output offset.

There is another possible error source which you need to check. Your scope offset controls may be contributing to the problem. Ground both inputs, and check to see what the scope is telling you the voltages are. Adjust the offsets to get exactly zero volts on both inputs, then recheck your measurements.