Roth IRA accounts have a "seasoning" rule, meaning the account has to have been in place for five years before you can take money out as a qualified distribution without paying a tax on the investment earnings withdrawn. Your current financial plan has a five-year horizon before you want to use money in the account for the down payment on a home, so that shouldn't be an issue. The first-time homebuyer distribution is limited to $10,000.
It's common for mutual funds to limit your trading in a fund, so you can't actively trade mutual funds in your retirement account, or any account. An exchange-traded fund, or ETF, may exist that closely matches a mutual fund. An ETF can be traded frequently, even during the trading day, which isn't possible in an open-end mutual fund.
I'm not aware of any penalties for excessive trading in a Roth IRA when that account is established as a brokerage account. Your Roth IRA brokerage account can't be a margin account where you can borrow any funds from your broker to invest. That keeps you from day-trading the account, but you can still actively trade the account.
Any taxes due on the investment earnings you take out of the account prior to age 59½, in general, are taxed as ordinary income. Investment earnings taken as nonqualified distributions would not be taxed as capital gains.
Recognizing capital losses in a Roth IRA account is possible, but you can only do that when all the amounts in all of your Roth IRA accounts have been distributed to you and the total distributions are less than your unrecovered costs from the accounts.
The typical retail investor won't be able to successfully trade a Roth IRA account and pick up a substantially higher return on the account than he or she would by investing the account, especially after considering the trading costs. Trying to swing for the fences in order to grow a few thousand dollars into a much larger balance over the next five years to finance the future down payment on a house isn't a very realistic plan.
That said, if you want to contribute a few thousand dollars to a Roth IRA brokerage account and see how you do with actively managing the investments in the account, it's a great way to learn about the stock market and investing.