I'm on my second playthrough, and I'd like to complete Lily's quest. I do know that you have to meet four criteria before you hear the recording of Lily's grandchildren, that then allows you to do the dialogues regarding Lily's medication.

From what I've read, you need to:
1) Talk to Lily about Leo- I did this when I initially met her
2) Talk to Lily about her psychotic breaks- I did this following a battle with giant Radscorpions
3) Sleep or wait- This part is confusing. I've done both, as well as simply just stand around without going to the 'wait' mode.
4) Talk to Lily about her medication- I don't think I've done this, and there aren't any dialogue options for it that I can see.

Does anybody have any suggestions, hints, tricks or tips about this?

My second question is regarding the "Whole Gang's Here" trophy. Now, I'm a bit of a perfectionist, and try and get 100% completion on what games I feel are feasible for this achievement. Now, this is a very easy trophy to get, IMO and I have recruited Arcade, Boone, Cass, Veronica, Raul, Lily, ED-E, and Rex. Now, I do know this trophy/achievement is bugged, and that all you apparently need to do is recruit any companion as many times as there are companions (8x, no?). I think I've done this, but no luck. Any other suggestions or tips regarding this trophy?

I had trouble at first, but if Lily has taken her medicine then you have to go through the option "Tell me about yourself" again and it should give you a new option telling you about her medicine. After you talk to her, wait or sleep and she'll start the recording and let you start her unmarked quest.

But that's the thing. I go to "Tell me about yourself" and see no option regarding medication. So, I'm thinking she isn't taking her meds, at least when I'm watching. Is there a way to get her to take her meds? I've been in several battles, and let her go all the way down towards 0 health, and have even had her go unconscious once. But still no medicine options.

Have you gone through all the options? I think the medicine option is hidden is dialouge option that you have already used. If she hasn't taken her medicine, she will take it randomly while traveling with her and can be noted when she says "time for your medicine, Leo" If she says that, you can talk about the medicine