Problems triggering opto-isolator with logic gate
High level output current [of the AND gate]: -400 uA. Low level output current: 8 mA
Yet, the opto-isolator desires a high level current of 5-15 mA and low level current of 0-250 uA.
I don't know how on earth I'm supposed to satisfy both of these current ratings,

There are 100's of different AND gates to choose from in the world. You're not stuck using 74LS08.
If you choose a CMOS gate instead of TTL, you will most likely be able to get the required 5 mA. If you need TTL input levels for your gate, you can use the HCT family which has TTL input levels but CMOS outputs.
If you can't find an AND gate that can provide 5 mA output while meeting your other requirements you can use a buffer gate or a transistor buffer between the AND gate output and the optocoupler input.
Or you could use a NAND gate and connect its output on the cathode side of the optocoupler LED instead of the anode side.