You're talking about a taxable brokerage account. Sure, lots of people do that.
If you're above the 15% taxable bracket, then you pay taxes on dividends and short/long term capital gains (cap gains only trigger if you sell off a stock/fund). If you're below that taxable bracket, then no, you don't pay anything on the dividends/cap gains
as long as you hold said funds for at least a year*. And yes, you can sell or buy at any time with no restrictions/fees other than what are built into the funds themselves or what your brokerage charges for buying and selling if any (check the funds you're buying - the expense ratios and any short-term redemption fees and the like).
So if you're investing in a taxable brokerage account, over the 15% tax bracket and aren't selling off your funds all the time, the only taxes you'd owe would be on the dividends generated by your funds. As this is a taxable account, you'd want to make sure you hold the most tax efficient funds possible in there.
http://www.bogleheads.org/wiki/Principles_of_tax-efficient_fund_placementThe proper order to invest according to account (buckets) if I am remembering correctly:
work-sponsored tax deferred account (401k/403b/other) - up to match (if any)
HSA if you have access to one - up to maximum ($3,350 for individuals/family coverage $6,650)
traditional IRA/Roth IRA (depends on which one works best for your situation) - up to maximum
($5,500 currently)work-sponsored accounts - up to maximum
($18,000 currently)taxable brokerage account
*edited to add the hold for a year part and correct order and maximums on accounts, got them reversed - gah!... thanks for the catch FrugalNacho!)