Can I trade stocks in my 401k mutual fund account?

doubtful. Most 401k's are limited to mutual funds only for very specific reasons. One of the main ones is that they are obligated to track and differentiate between the gains/losses/interest/dividends etc. They are also obligated to ensure that every account is invested only in investments allowable by law. Additionally they are required to ensure that no money leaves the plan unless it's allowable and proper procedures take place. Brokerage accounts make this tracking difficult as brokers are not generally concerned with this aspect of the plans...they just want the transaction fees. Those plans that do offer this option is usually more expensive on both a plan level and an individual level. Often times the added expense make it less than worthwhile.

That being said....99% of the country should not be trading in individual stocks anyways. The 401k is not a play thing...it's a retirement investment vehicle. While risk is inevitable, it should be managed and minimized to ensure a safe retirement. Investing in individual stocks before you have an adequate retirement balance (> $200,000) means you aren't diversifying your account enough and your risk will be out of proportion to your return. I certainly have no problem with people having the right to trade stocks in a 401k...but most should not exercise that right.

If your 401(k) is invested in mutual funds, no, you cannot "trade stocks." If your 401(k) plan is set up as a self-directed brokerage account, then you can. It all depends on your company's plan.

It really just depends on your company. Most larger companies will only have company stock and mutual funds as options. Most will only let you move in and out the funds a certain number of times per year.
Smaller plans administered more like an old SEP might let you trade more often. Talk to your plan administrator.