For someone in the USA, is it possible to have:
1) A desktop or online based wallet if you don't have a photo ID.
2) Ability to buy btc without needing to provide ID.
That is scary to someone who has had identity stolen before.
I agree, identity theft is a real concern. I don't trust many of the bitcoin exchanges to store my data and prevent it from being stolen, so I'm very careful about how I approach services that enforce AML/KYC (anti money laundering and know-your-customer) policies.
On the first point: With the Bitcoin protocol itself (your wallet), you never need to provide ID. It's completely decentralized and peer-to-peer. No government intrusion.
On the second point: This is a bit trickier. You can buy BTC on LocalBitcoins, but it's getting harder and harder to buy with cash and without ID. In places like the US and western Europe, regulators and police are pressuring high volume LBC traders to act in compliance with money transmitter laws. So they've started asking for ID.
This is one reason why I really like to earn BTC and hold it, wherever possible. I hate forking over my documents.....