Unfortunately, I'm doing a half-assed reading job, so giving half-assed answers :-). My apologies.
For question 2, the whole thing is actually in my answer. It's in the Equity side of the equation. Revenue - Expenses = Profit. The inventory and asset side doesn't really have anything to do with profits per se; sure those profits may be cash assets in the cash accounts... but so might cash from a loan or similar non-revenue source. So the income statement is based on those bookings. The cash, the inventory, etc. on the Asset side of the equation don't really have much to say about profit. It's just stuff you own. The reason it's in the breakout of the Shareholder Equity side of the equation is because it's the shareholders who are exposed to profit or loss; the concept of profit/loss just doesn't need to appear elsewhere. Often times you'll hear accounts talk about "income statement accounts" and "balance sheet accounts". Revenue and Expense accounts are the "income statement accounts". At the end of a fiscal year, the income statement accounts, which state the profit or loss, are basically netted out and booked as a "Retained Earnings" entry. The "balance sheet accounts" including asset accounts like cash and inventory valuation or liabilities, on the other hand carry their values year over year.
To your inventory valuation question (you may ultimately be sorry you asked :-) ). Inventory valuation accounting is more complicated than it first appears and how it happens can best be summed up as: it depends on a number of factors. The nature of the business, the preferences of the accounting management team, etc. Sometimes there can be multiple inventory valuation methodologies in operation at once depending on who is consuming the information being produced or the specific inventory being valued. Note that with all methods, the basics from my first comment will still apply: these different methods are primarily aimed at how to arrive at a unit cost
First, a couple disclaimers. I'm not a CPA/Accountant/etc. I am principally a technologist and some of what I'm going to be giving you is based on the understanding I've developed for my own purposes of the issues rather than something more textbook. I do have around 30 years of experience which consists of a combination of management roles within companies with significant and complex inventory management accounting (think chain retailers), ERP implementation consulting into retail, distribution, and manufacturing companies including working with finance teams for accounting implementation, designing a Retail Method stock ledger for an ERP system, and finally redesigning the costing mechanisms of a manufacturing-centric ERP system (as well as doing the development work) along with a fair amount of research into these questions. None of that promises that I know what I'm talking about, but at least I've had to think about the kinds of questions you are asking at some depth.
Inventory valuation. There are a number of different approaches as others now in this thread have mentioned. First In-First Out, Last In-First Out, Weighted Average Costing (more on this in a bit), Standard Costing, Actual Costing, Job Costing, and Retail Method, to name a few. These methods have different histories and traditions and some are more prevalent in certain industries than others.
Knowing which to use involves knowing which goals you're pursuing. First there's a division at large in accounting which needs to be understood. There are two kinds of accounting of interest here: Financial Accounting and Management Accounting. Financial Accounting is about producing financial statements for investors, banks, etc. and for tax reporting (a whole other can of worms I will mercifully/largely avoid here). Then there's Management Accounting. Management Accounting is concerned with internal nuanced understanding the operations of the business at large. These goals don't always see the world the same way and will have different demands of accounting. You'll see terms like GAAP ("Generally Accepted Accounting Principles", the U.S. standard for Financial Reporting) and IFRS ("International Financial Reporting Standards", same as GAAP for much of the rest of the world). These are targeted at investor financial statements and so fall into the Financial Accounting category.
Costing methods associated with Financial Accounting (GAAP/IFRS) are Weighted Average Costing (conditional for tax purposes), FIFO, LIFO (only under GAAP, not IFRS, allowed by the IRS under certain conditions), Job Costing (conditional), Actual Costing, and I believe Retail Method is also accepted in GAAP/IFRS, but not tax (I could be wrong), but I really see it as a Management Accounting method. Management Accounting methods include Standard Costing and Retail Method. Note that for the Retail Method especially and Standard Costing to a lesser degree, their usage has changed over time from use in Financial Accounting to be more Management Accounting focused. Also know that many applications implementing costing conflate these purposes, especially on the lower end of the market: they rely on expertise in the accounting area to produce correct reports or laxity by investors and taxing authorities to care. Bigger systems tend to get it and implement things right (if annoyingly so).
Of all of these, for Financial Accounting, the most popular would be a flavor of weighted average costing or FIFO. Others mentioned LIFO as being more accepted, but that's backwards: it's FIFO. LIFO is not acceptable under IFRS and really only the IRS acceptance (and interestingly the tax advantages that can show up with LIFO) that keep it alive at all.
-- Weighted Average Costing. As you've suggested, when you have inventory where individual units stocked are fungible with all other units of the same item (I call this "Unit General Costing"), you can simply average the receipts of inventory with that already on hand and arrive at an average cost. There are different flavors of this basic approach. What I'll call "perpetual average costing" involves immediate updates to the average unit cost whenever new units are received into inventory; simply put: (new receipt value + existing inventory value) / (new receipt quantity + existing on hand quantity) = Current Unit Cost for things like sales transactions. Note that transactions like sales shipments don't change the current unit cost, only receipts of new inventory such as new purchases. There's also what I call "periodic average cost". In periodic average cost methods, the cost is calculated at the beginning of some period of time: a day, a week, maybe even a fiscal month. The same cost calculated for that period is used in all transactions that issue inventory (e.g. sales order shipments), when the period rolls over to the next, the current inventory with the actual receipts of the period are then used to calculate a new average cost for the new period.
-- First In/First Out. The basic mechanics of this costing method is pretty evident from the name. You keep a log of received inventory quantities and the costs they were obtained at. As you issue (ship) units you relieve the inventory at unit cost with the oldest inventory going first using the costs from the log. Sure you might not be shipping the specific units in FIFO order, but who cares? If the inventory is truly in that "unit general" accounting category... it doesn't matter.
Naturally there are conditions that crop up that can foil things. For example, there are cases where taking your on-hand inventory quantities into negative territory is valid; also, there are companies that have legitimate reasons for backdating transactions. Think what that does in either of the Average Costing or FIFO costing cases :-). And we haven't even discussed loading the costs that you find out about later for things like capitalized freight, etc. There are ways of dealing with these things, but the common-case gist of it all I think is captured above. (plus I'm a couple or four beers in now and should probably stop :-) ).
I don't know if that provides clarity, but at least it should give you some sense of the size of the question you're asking.