Can you place a buy limit order above the current share price (or place a sell limit order below the current share price vice versa)?

I think what you are looking for is a Stop order or Stop limit order.

Not every brokerage has this. I use Thinkorswim so I know I can.

It depends if your broker allows it. Many only allow limit and market orders which wont allow this since they'll fill immediately (since the buy price current price or sell price < current price).

However, if your broker allows, you can use a stop limit/stop market/trailing stop order to sell under the current price (eg if you're trading to cut loss). Alternatively, for the buy side, you can use a market if touched order to buy above a certain buy price (eg if the stock experiences a breakout and you want to get in quickly). IBKR supports all of these order types but Im not too sure about others.​​​