I'm thinking of creating a site where people can deposit $1 into their account. I then invest it for them into an expertly picked penny stock. Is this legal to do? What problems would be faced doing this?

You would need to be licensed to be able to handle other people's investments.  Start with that before you go anywhere else.

One problem is quite obvious: "Expert-picked penny stock" is an oxymoron.

Just an unbelievably stupid idea, to be very blunt. (Don't worry we all have these) 1. By you saying, "expertly picked penny stock" it makes me think you just stumbled upon a pump and dump site and fell for their terrible ploy to pull in the most gullible of internet users. Basically, you probably don't understand how to trade, or as you say "invest", in penny stocks anyways. 2. Essentially, this is a hedge fund. The legal side to forming a hedge fund is far too complicated for you to get sorted out for something as small as an idea as this.

This sounds of questionable legality.  First off you have to be licensed as an investment adviser and fiduciary agent.  Second you have to word the service you offer so that it is not some kind of a Ponzi Scheme.  Doing funny business on the Internet maybe an express ride to the nearest jail cell.

If it's random people like that you definitely need to set up a legal structure. I believe Citi or JPM or one of the big banks has a guide floating around. You would also need to be licensed. But please don't do this. It's a horrible idea on so many levels. Just don't.

Sounds like you are trying to create a mutual fund or investment trust. I hope that you have the requisite licenses, if not, I wish you a pleasant time in prison. Based on your question, it seems that you do are not licensed and that is the first thing you need to do. Once you do that, you'll be have more knowledge and be able to proceed.

Oh my god !!! You have no idea what you are saying or thinking. Look up the jobs act with regards to crowd funding and set up some kind of donation fund and let people know they are contributing a donation and will not expect their money back or any reward for it. Try being honest and straight forward let people know you are experimenting so that you can learn some thing but the minute you utter the word invest and add the phase invest for others your into a whole world you know nothing about it and penny stocks? God!! for now you have terrible judgement. I hope you have not already begun to do this and are just asking for opinion and advise, your getting it. Forget all about your idea and work to gain some knowledge and if you ever think you have enough knowledge to do what you propose legally consult a securities attorney first. See you in 10 years and hopefully you have not given up on you idea of learning about investing and the securities laws both federal and state. Stay with it you may make a career out of this yet. I encourage you to learn about investing and about securities laws and maybe someday you will get that licensed and be qualified maybe as a stock broker, a proprietary trader, an investment adviser , a securities lawyer or some related profession but for now  Hell NO !!!!!!!!!!!!