If my mutual fund's minimum investment is $10K, does that mean I can sell shares after a year and fall below $10K without penalty?
"Minimum investment" is what you need to put into the account to open it.
They may charge you maintenance fees if you fall below some amount, but once you're in you're in. They'll be more than happy to take their management fee. :)
Read the prospectus. It should tell you if there's a minimum balance requirement, and if so, what the fees/penalties are for falling below that requirement.
Not always. The fund may have a minimum requirement that lets the value falling below the minimum due to the market have no impact, but they may not allow withdrawals if you fail to meet the balance. Otherwise, you could open the account for $10K and withdraw $9000 the next day.