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2-way pegging means that value may be transferred at a pegged (fixed) rate in either direction.

This special form output is a script which is able to understand an embedded proof-of-spend from another chain, which validates the accounting rules (you need to spend X bitcoins to claim X sidecoins, and vice versa), and which makes sure that claimed coins go to the indicated recipient.

this question related to the red-marked quote:

What happens if I moved a BTC to a side chain to be used with a financial instruments such as options or even betting and GAINED an equivalent value to what I moved out. I now have 2 sidecoins but previously moved only 1BTC. can I go 1-way with one of them?

Yes, because the 1BTC that you gained through trading had to be originally moved into the sidechain as well, and it can be redeemed in the same way.

Where did this gain, this additional sidecoin, come from?

Who knows? Trading? Betting? SatoshiSidecoinDice? Either way, that coin didn't come from nowhere. It came from someone who moved it into the sidechain from the mainchain.

So is it the case that at any time, anyone can move any amount of BTC that they control into any sidechain?
EDIT: What I am getting at is whether there are limits to the number of units on a particular sidechain. One does not need to find a "seller" or "redeemer" of sidecoins to make a BTC-sidecoin trade, since the sidecoin is not truly a separate unit of currency, so one is not really trading into it, correct?

(Thank you all for your patience and generosity in responding to my questions. I assume others trying to learn, too. I'm trying to piece things together from here and the two reddit threads.)

This is the critical question I'm still confused about. If 1 sidechain BTC always = 1 mainnet BTC, then how will sidechains be useful for assets? They'll be fine for things like fast BTC transactions etc.
Example: my sidechain asset (gold or whatever) that I paid for with 1 sidechainBTC is now worth 2 sidechainBTC. Do I redeem it for 2 sidechainBTC, then re-peg back to mainnet BTC?

Yes.

My understanding of at least part of the proposal is that the side-chain will allow Bitcoin devs to introduce new/experimental features, such as Counterparty-like features, without disturbing the mainnet or requiring multiple hard-forks/upgrades of Bitcoin. Over time, once the new/experimental sidechains become fully vetted, it may happen that the sidechain becomes the mainchain. In this way, a "Bitcoin 2.0" could be introduced gradually.

The asset-issuance questions apply equally to Counterparty, Mastercoin, Colored Coins, etc. A certain amount of sidechain-BTC will be required in order to create an asset. This asset may be traded for BTC just as Counterparty-assets (which, remember, are traded using BTC as the carrier, so they are actually BTC-assets in some way) may be traded for Counterparty-as-currency or BTC-as-currency.

You will never need to "re-peg". The peg simply is. It's built into both the sidechain-BTC and mainnet-BTC. If you have 1 sidechain BTC, it will always be redeemable for 1 mainnet BTC directly through the transfer process. If you have 1 mainnet BTC, it will always be redeemable for 1 sidechain BTC in the same way. This is simply how the system will be built, and it cannot be changed. If somebody did try to change it, their transaction would not be recognized by the honest nodes in the Bitcoin system.

Perhaps the word "peg" is wrong, but I don't know a better word that communicates the idea. Simply "equivalent"? It is not a "peg" like with fiat currencies where one currency is manipulated to always match another. It is the same currency. It is like if you have 1 USD as a dollar bill, and you deposit it into your bank account. Then you no longer have a dollar bill, but you have 1 USD in your bank account. This can be changed back into a dollar bill by going to your bank and withdrawing it. You can trade back and forth infinitely PaperUSD <==> BankUSD, and the bank is never going to try to give you more or less than 1 BankUSD for 1 PaperUSD.