Towards your question #1 The basics of double-entry accounting run off of the equation that (Assets) - (Liabilities) = (Owner's Equity).
In terms of double entry bookkeeping, you always need 1 "event"'s set of transactions to balance the left (A - L) and right (OE) sides of the equation. Here are some examples to show what I mean:
Ex 1: Company starts with $100 in cash from last period. This would show up on the balance sheets as A: $100 Cash L: $0 = OE: $100.
Company buys $50 worth of ice cream. This would record as A: -$50 Cash --> A: +$50 Inventory
Company sells all the ice cream for $75. This would record as A: +$75 Cash --> A: -$50 Inventory --> OE: $25 (Profit).
If this was the only activity for the period, then at the end of the... month? The accounts would show:
Assets:
Starting: $100
-$50 Inventory Purchased
+$75 Revenue on Sales Cash
Ending: $125
Starting: $0 (Assuming we had none to start)
+$50 Inventory Purchased
-$50 Inventory Sold
$0 Ending Inventory
Liabilities: nothing
===
Owner's Equity:
Starting $100
+$25 Profit on Sales
Ending: $125
Hopefully this helps you to see how you're really recording each event in two (or more) places, hence the 'double entry' aspect of the name.
Ex 2: This gets at your specific question, which is something like 'How do I record a revenue transaction that doesn't really have any associated inventory or costs?'
Remember that your "A - L == OE" equation has to balance, so you either need to simultaneously (a) increase and decrease Assets... like buying ice cream/inventory, (b) increase Assets and increase Liabilities, (usually by 'buying' inventory and setting up a short-term liability) or (c) increase Assets and increase Owner's Equity.
Based on what you've got in your example, I would expect the lottery winnings/inheritance stuff to show up under "Assets" in some kind of setup like "Assets... Cash: +$250,000 Lottery Winnings" --> "Owner's Equity... +$250,000". If you wanted it to be in an account "savings" after that, I would personally record that as "Assets... Cash: -$250,000 Transfer to Savings" --> "Assets... Savings: +$250,000".
Hopefully that shows you exactly how revenue... or income passes through from accounts under "Assets" into overall "Equity".
Now, on your second question: no, you've totally screwed it up. But at the same time, it seems like this is an attempt at tracking your personal net worth in accounting software, so technically there's no "right" or "wrong" way. I'm not personally familiar with GnuCash, so I can't give you a walk-through on that... maybe someone else can.
What I can do is tell you the accounts that I would probably set up in GnuCash or Quickbooks or Peachtree or whatever accounting system you were to use. I'm also assuming that this is a personal accounts setup, but I'll be glad to edit it to reflect your real situation if you comment.
That being said, I would have the structure be something like:
Assets:
Cash on hand
Checking
Savings
MMA or CDs (Short-term investments)
401K/IRA/Whatever (Longer-term investments)
Real Property Assets (House, Car, Computer, Bike)
Liabilities:
===
"Owner's" Equity: (you're welcome to get as specific on this as you like, but that's just extra work...)
From there you use double entry book-keeping with a vengeance. For the 'right' way to do your two transactions, you would have some kind of 'revenue' hit one of your "Assets" accounts (maybe +$250,000 to Savings?) and record a corresponding increase to your Net Worth (maybe +$250,000 to Overall Net Worth?). Then you would do the same thing for your Inheritance: (+$500,000 to Savings?... +$500,000 to Overall Net Worth?).
We can talk about how you would account for taxes and other expenses some other time... that's definitely a topic for "Lesson 2". :)