Yes, the iPhone 8 Plus supports Qi wireless charging. This model features a glass back specifically designed for wireless charging.
Whether or not an iPhone will charge through a case depends. For the most part the answer is yes. Most companies that design iPhone cases now take wireless charging into account. The thinner a cover is, the more likely it is to receive charge through the case. The thicker and more decorated the case is, the less likely that it will receive charge.
You’ll also want to take into account the quality of the phone charger that you purchased. If it’s a cheap $4.00 charger that you grabbed online, it likely won’t be as effective or as powerful as some of the other premium brands that are available.