I recently moved out from my parents place, after having built up sufficient funds, and gone through these questions myself. I live near Louisville, KY which has a significant effect on my income, cost of living, and cost of housing. Factor that into your decisions.
To answer your questions in order:
When do I know that I'm financially stable to move out?
When you have enough money set aside for all projected expenses for 3-6 months and an emergency fund of 4-10K, depending on how large a safety net you want or need. Note that part of the reason for the emergency fund is as a buffer for the things you won't realize you need until you move out, such as pots or chairs. It also covers things being more expensive than anticipated.
Should I wait until both my emergency fund is at least 6 months of pay
and my loans in my parents' names is paid off (to free up money)?
6 months of pay is not a good measuring stick. Use months of expenses instead. In general, student loans are a small enough cost per month that you just need to factor them into your costs.
When should I factor in the newer car investment? How much should I
have set aside for the car?
Do the car while you are living at home. This allows you to put more than the minimum payment down each month, and you can get ahead. That looks good on your credit, and allows refinancing later for a lower minimum payment when you move out. Finally, it gives you a "sense" of the monthly cost while you still have leeway to adjust things.
Depending on new/used status of the car, set aside around 3-5K for a down payment. That gives you a decent rate, without too much haggling trouble.
Should I get an apartment for a couple years before looking for my own
Not unless you want the flexibility of an apartment. In general, living at home is cheaper. If you intend to eventually buy property in the same area, an apartment is throwing money away. If you want to move every few years, an apartment can, depending on the lease, give you that.
How much should I set aside for either investment (apartment vs
10-20K for a down payment, if you live around Louisville, KY. Be very choosy about the price of your house and this gives you the best of everything. The biggest mistake you can make is trying to get into a place too "early". Banks pay attention to the down payment for a good reason. It indicates commitment, care, and an ability to go the distance. In general, a mortgage is 30 years. You won't pay it off for a long time, so plan for that.
Is there anything else I should be doing/taking advantage of with my
money during this "living at home" period before I finally leave the
If there is something you want, now's the time to get it. You can make snap purchases on furniture/motorcycles/games and not hurt yourself. Take vacations, since there is room in the budget. If you've thought about moving to a different state for work, travel there for a weekend/week and see if you even like the place.
Look for deals on things you'll need when you move out. Utensils, towels, brooms, furniture, and so forth can be bought cheaply, and you can get quality, but it takes time to find these deals.
Pick up activities with monthly expenses. Boxing, dancing, gym memberships, hackerspaces and so forth become much more difficult to fit into the budget later. They also give you a better credit rating for a recurring expense, and allow you to get a "feel" for how things like a monthly utility bill will work.
Finally, get involved in various investments. A 401k is only the start, so look at penny stocks, indexed funds, ETFs or other things to diversify with. Check out local businesses, or start something on the side. Experiment, and have fun.