What speed does the home network need to be so that Air Play is the same as connected by a cable?
No, you can never guarantee the network connectivity. You could always experience interference from other networks, jammers, etc. which temporarily block signal reception. Similarly with a cable you could have defective cables that aren't working perfectly.

In general you can achieve network speeds with 802.11ac that well exceed the speed of a 100 Mbps cable connection and approaches the speed of a 1000 Mbps cable connection (or even exceeds it in optimal scenarios).

If your question is whether or not AirPlay from a MacBook to an AppleTV can work well over 802.11ac - then the answer is yes. However, you'll have to try yourself with your own network components and surroundings to determine if it works well for you - there's no guarantee. Usually it works without any problems.

You can use Activity Monitor to monitor the actual amount of transferred data. You can also hold down the Alt key while clicking the WiFi icon in the menu bar on your MacBook to see extra statistics, including the current estimated transmission rate.