Nope. Your still using hotel wifi which is usually slow so it will still lag. The only way it wasn't slow was using my mifi but it's limited data so I'm not going to use it and recently, i just canceled it since my work gave me one as well. But I never use my work one for personal so I haven't hooked up my Chromecast in a while to it.I'm in the same boat. Last trip I was able to setup Chromecast to hotel network. But casting an hd movie resulted in horrible lag. Just didn't work.
Therefore, I'm considering buying a travel router from Amazon and test the hd quality. If that won't work, I'll be force to take a 11" laptop with me with hdmi port.
No it doesn't and it's absolutely unnecessary. If you want to connect to your TV just cast to it. A Chromecast will allow you to cast anything to your TV, basically the TV mirrors the phone screen. If you have a smart TV you don't even need a Chromecast, you can cast directly to it with Allcast and apps like Netflix and Youtube will couple to the TV's Netflix and Youtube apps allowing you to control them with the phone but playing through the TVs Ethernet or WiFI connection.
Chromecast still needs am internet connection.
Anyone that travels know how hotel wifis are managed and how to log onto them. Can't do it with Chromecast. In this case, HDMI cable is needed.