1) Does this Ethereum wallet accept ETH payments? (I saw something about ETC vs. ETH and I care about ETH)
Yes it does, however it requires you to sync the blockchain before you can view balances.
2) What all does one need to send a payment? Just the user's Address hex?
Yes. In order to send ETH from one address to another, a sender needs the receiver's public address (which looks something like this:
0x4bbeEB066eD09B7AEd07bF39EEe0460DFa261520) and enough ETH to send the amount being sent + the cost of ETH used for gas.
3) When a payment is sent, how long does it typically take to reflect in the receiver's wallet?
This depends on how the gas is configured by the user. Can be as fast as 15 seconds or as slow as forever. If you're using the Ethereum Wallet app it may take a long time to notice the changes if it isn't synced up, because the only way the Ethereum Wallet can see the changes are if it's up to date with the block number that the transaction was approved in.
4) Is there any risk of losing Ether funds if the wallet is unable or doesn't sync with the Network Node for say 1 week?
No. The way that the network works is that the funds are allocated to a public address (which is only accessible via the private key for that address) whenever funds are sent to that address. The funds don't exist in a wallet. They only exist on the blockchain. The wallet is just a software that generates and/or stores your private key which is how you "prove ownership" of the public address. The private key IS ownership of the public address. Funds will sit there (showing as in the public address of the receiver) until someone with a private key for that public address accesses it and sends it away to a different address.
5) I understand with Ethereum Wallet there is a password (user-created) and keystore, but isn't there also supposed to be a private key to save/backup somewhere?
The password in the Ethereum Wallet is used to encrypt the private key into the keystore file. The keystore file IS your encrypted private key. Using just a private key to access your account is considered insecure for most users. Because private keys ARE ownership of a public address, private key management is of the utmost importance in cryptocurrencies. Normally, people in the space would recommend purchasing a hardware wallet (such as a Ledger Nano S or a Trezor) for this reason. Hardware wallets don't require you to know your private key and they don't require you to expose your private key to the internet in order to interact with your Ethereum keypairs.
6) I'm confused about Ethereum ICO and something about ERC20 or other? I'm not really sure, but I just want to make sure I can accept ETH payments without issue using Ethereum Wallet 0.11.1.
ERC20 tokens are implementations of Ethereum code to generate a token with fixed supply. They don't really have anything to do if you just want to accept ETH payments. There is a bit of background reading for them, but you can start here if you're interested: https://cointelegraph.com/explained/erc-20-tokens-explained
I would recommend using a different software due to the complexities of key management and the current relative difficulty of using the standard Ethereum Wallet software and keeping your node up-to-date. Maybe look into MyCrypto or MyEtherWallet. And definitely, at least look into the purchase of a hardware wallet if you're planning to store more than a few hundred dollars worth of ETH/ERC20 tokens. The cryptocurrency ecosystem is rife with scams that target both individuals and vast swathes of cryptocurrency users, so I can't overstate how important it is that you read up on how to secure your funds.