Interesting, I hadn't thought of that. So I decided to do some research, and guess what I found? I looked at some words for "corpse" and "труп" is inanimate. But "мертвец" is animate...
The answer to your question about "друг", I am fairly positive, is that the definition of "animate" or otherwise is attached to the noun and not the individual. So "друга" would remain "друга".
You may have misspoken - verbs are conjugated based on the noun and there are no cases associated with them. Nouns, pronouns, and adjectives are declined into cases like accusative, genetive, etc.
I go back and forth on whether I want all of that information on the hints in DL. Just a gender and a case would be good for me. There are a lot of excellent options to find the right forms.
Morphological Analysis allows you to enter in any form of a word and see all of the forms of the word. It can help you determine which case something is in.
Wiktionary is also extremely helpful both for declensions and verb conjugations.
I highly reccommend you start to make yourself a chart of the conjugations and declensions in a format that makes sense for you. It takes time to memorize it - making a chart reinforces what you know (and highlights what you don't) and since it's yours, it's organized in a way that's ideal for you.
Verbs don't have accusative and genitive forms. In general they match up pretty well with English; they take a direct object in accusative case and/or an indirect object in dative case. There are only a few where we would expect accusative case in English but in Russian it's dative.
It may be that your confusion is about for which nouns the accusative form matches the genitive. Well, that's really pretty easy. If it's a person or an animal, dog, turtle, electrician, anything that you would expect to be animate, it's animate in Russian. Anything you wouldn't expect to be animate, table, grass, is inanimate. Things like trains are inanimate because they aren't living. There are only a handful of odd words (like мертвец above) that have to be learned and none are important.
In this case "друга" is simply the genitive case of "друг" because "потерял" requires the genitive. The feminine version of "друг" is "подруга" but both only mean a platonic friend.
There's no separate word for a "boyfriend"/"girlfriend" in Russian. People usually just use the words for "guy" ("парень") and "girl" ("девушка") with a possessive. It's so ubiquitous, that "моя девушка" would invariably be understood as "my girlfriend" and that's how this course presents expressions like this (though that confuses some learners, because they end up thinking that "девушка" always means "girlfriend", which it doesn't).
To answer your question, "How did you lose your girlfriend?" would be "Как ты потерял свою девушку?".
Technically in some instances "подруга" can be translated as "girlfriend" but in most cases it would be misleading. It's possible to use "моя подруга" euphemistically for a "girl I'm dating", but that's just not the actual meaning of the word. Also it's possible to interpret "my girlfriend" as "my platonic female friend" but that meaning is not the first thing anyone would think of when they hear it.