So can a pro-se litigant successfully do discovery on her own?
Yes. You can serve discovery requests on your own, and can file a Motion to Compel if they do not respond.
Do I need to depose my ex to successfully get what I need?
That's very case specific. It depends on what you need. Depositions are not done in every case. If he's not claiming inability to pay and you have no reason to believe that he is hiding assets, it may not be necessary. But you can do a deposition if you wish.
Can I do an informal discovery such as a request for documents?
Yes. You can also do Interrogatories and Requests for Admission, or a Deposition by Written Question (if you didn't want to do the question/answer format of a normal deposition).
Can I use something like his past (and I'm sure also present) use of alcohol and marijuana when he's partying with his buddies, and once he came on a Sunday morning to take care of the girls hung over?
If he is requesting custody, absolutely. That is very relevant to his ability to parent. Generally, frequent use of alcohol can count against a parent, as can any use of drugs, including marijuana.
How much can I use how he cares for the children against him? For instance, he still changes our 8 and 5 year old girls, and they run around naked in front of him. And one of the girls saw him walking around naked in the bedroom/bathroom as well. These are concerns, but how much does that weigh on the case?
Those things are pretty concerning and absolutely can be brought to the judge's attention. It's up to the judge to determine how much weight to grant any specific factor, but essentially, you're trying to create a wall of reasons that your ex shouldn't have custody. All of these things, and the partying/drug use are bricks on that wall. The more information you have, the better.
Is there a website I can go to for all the required forms to request documents (bank statements, paystubs, bonuses, etc)? I just don't know that since I'm not represented, if I can do that on my own. Please help. I appreciate your thoroughness above.
There are websites with examples. In any given case, those examples would usually be tailored to make sure that they specifically request the information that you want. Popular sites are www.uslegalforms.com and www.legalzoom.com. A law library would also have books that you can look at and use to type up your own requests.