Yes. There are several downsides to this strategy:
You aren't taking into account commissions. If you pay $5 each time you buy or sell a stock, you may greatly reduce or even eliminate any possible gains you would make from trading such small amounts. This next point sounds obvious, but remember that you pay a commission on every trade regardless of profit, so every trade you make that you make at a loss also costs you commissions. Even if you make trades that are profitable more often than not, if you make quite a few trades with small amounts like this, your commissions may eat away all of your profits. Commissions represent a fixed cost, so their effect on your gains decreases proportionally with the amount of money you place at risk in each trade.
Since you're in the US, you're required to follow the SEC rules on pattern day trading. From that link, "FINRA rules define a “pattern day trader” as any customer who executes four or more “day trades” within five business days, provided that the number of day trades represents more than six percent of the customer’s total trades in the margin account for that same five business day period." If you trip this rule, you'll be required to maintain $25,000 in a margin brokerage account. If you can't maintain the balance, your account will be locked.
Don't forget about capital gains taxes. Since you're holding these securities for less than a year, your gains will be taxed at your ordinary income tax rates. You can deduct your capital losses too (assuming you don't repurchase the same security within 30 days, because in that case, the wash sale rule prevents you from deducting the loss), but it's important to think about gains and losses in real terms, not nominal terms. The story is different if you make these trades in a tax-sheltered account like an IRA, but the other problems still apply.
You're implicitly assuming that the stock's prices are skewed in the positive direction. Remember that you have limit orders placed at the upper and lower bounds of the range, so if the stock price decreases before it increases, your limit order at the lower bound will be triggered and you'll trade at a loss. If you're hoping to make a profit through buying low and selling high, you want a stock that hits its upper bound before hitting the lower bound the majority of the time. Unless you have data analysis (not just your intuition or a pattern you've talked yourself into from looking at a chart) to back this up, you're essentially gambling that more often than not, the stock price will increase before it decreases.
It's dangerous to use any strategy that you haven't backtested extensively. Find several months or years of historical data, either intra-day or daily data, depending on the time frame you're using to trade, and simulate your strategy exactly. This helps you determine the potential profitability of your strategy, and it also forces you to decide on a plan for precisely when you want to invest. Do you invest as soon as the stock trades in a range (which algorithms can determine far better than intuition)? It also helps you figure out how to manage your risk and how much loss you're willing to accept. For risk management, using limit orders is a start, but see my point above about positively skewed prices. Limit orders aren't enough.
In general, if an active investment strategy seems like a "no-brainer" or too good to be true, it's probably not viable. In general, as a retail investor, it's foolish to assume that no one else has thought of your simple active strategy to make easy money. I can promise you that someone has thought of it. Trading firms have quantitative researchers that are paid to think of and implement trading strategies all the time.
If it's viable at any scale, they'll probably already have utilized it and arbitraged away the potential for small traders to make significant gains.
Trust me, you're not the first person who thought of using limit orders to make "easy money" off volatile stocks. The fact that you're asking here and doing research before implementing this strategy, however, means that you're on the right track. It's always wise to research a strategy extensively before deploying it in the wild.
To answer the question in your title, since it could be interpreted a little differently than the body of the question: No, there's nothing wrong with investing in volatile stocks, indexes, etc. I certainly do, and I'm sure many others on this site do as well. It's not the investing that gets you into trouble and costs you a lot of money; it's the rapid buying and selling and attempting to time the market that proves costly, which is what you're doing when you implicitly bet that the distribution of the stock's prices is positively skewed.
Update (In response to your edit:)
To address the commission fee problem, assuming a fee of $8 per trade ... and a minimum of $100 profit per sale
Commissions aren't your only problem, and counting on $100 profit per sale is a significant assumption. Look at point #4 above. Through your use of limit orders, you're making the implicit assumption that, more often than not, the price will trigger your upper limit order before your lower limit order.
Here's a simple example; let's assume you have limit orders placed at +2 and -2 of your purchase price, and that triggering the limit order at +2 earns you $100 profit, while triggering the limit order at -2 incurs a loss of $100. Assume your commission is $5 on each trade.
If your upper limit order is triggered, you earn a profit of 100 - 10 = 90, then set up the same set of limit orders again. If your lower limit order is triggered this time, you incur a loss of 100 + 10 = 110, so your net gain is 90 - 110 = -20. This is a perfect example of why, when taking into account transaction costs, even strategies that at first glance seem profitable mathematically can actually fail.
If you set up the same situation again and incur a loss again (100 + 10 = 110), you're now down -20 - 110 = -130. To make a profit, you need to make two profitable trades, without incurring further losses.
This is why point #4 is so important. Whenever you trade, it's critical to completely understand the risk you're taking and the bet you're actually making, not just the bet you think you're making.
Also, according to my "algorithm" a sale only takes place once the stock rises by 1 or 2 points; otherwise the stock is held until it does.
Does this mean you've removed the lower limit order? If yes, then you expose yourself to downside risk. What if the stock has traded within a range, then suddenly starts declining because of bad earnings reports or systemic risks (to name a few)? If you haven't removed the lower limit order, then point #4 still stands.
However, I never specified that the trades have to be done within the same day.
Let the investor open up 5 brokerage accounts at 5 different firms (for safeguarding against being labeled a "Pattern Day Trader").
Each account may only hold 1 security at any time, for the span of 1 business week.
How do you control how long the security is held? You're using limit orders, which will be triggered when the stock price hits a certain level, regardless of when that happens. Maybe that will happen within a week, or maybe it will happen within the same day. Once again, the bet you're actually making is different from the bet you think you're making.
Can you provide some algorithms or methods that do work for generating some extra cash on the side, aside from purchasing S&P 500 type index funds and waiting?
When I purchase index funds, it's not to generate extra liquid cash on the side. I don't invest nearly enough to be able to purchase an index fund and earn substantial dividends.
I don't want to get into any specific strategies because I'm not in the business of making investment recommendations, and I don't want to start. Furthermore, I don't think explicit investment recommendations are welcome here (unless it's describing why something is a bad idea), and I agree with that policy. I will make a couple of points, however.
Understand your goals. Are you investing for retirement or a shorter horizon, e.g. some side income? You seem to know this already, but I include it for future readers.
If a strategy seems too good to be true, it probably is.
Educate yourself before designing a strategy. Research fundamental analysis, different types of orders (e.g., so you fully understand that you don't have control over when limit orders are executed), different sectors of the market if that's where your interests lie, etc. Personally, I find some sectors fascinating, so researching them thoroughly allows me to make informed investment decisions as well as learn about something that interests me.
Understand your limits. How much money are you willing to risk and possibly lose? Do you have a risk management strategy in place to prevent unexpected losses? What are the costs of the risk management itself?
Backtest, backtest, backtest. Ideally your backtesting and simulating should be identical to actual market conditions and incorporate all transaction costs and a wide range of historical data.
Get other opinions. Evaluate those opinions with the same critical eye as I and others have evaluated your proposed strategy.