From 1966, it's The Cyrkle with "Red Rubber Ball," a tune which reached #2 on the charts that year. I'm looking over the list of The Cyrkle's other records and the only one I recall ever hearing was "Turn-Down Day," which did not make my mixtape. "Red Rubber Ball" was written by Paul Simon of Simon & Garfunkel and Bruce Woodley of The Seekers and I recall liking the tune although I never saw a sun in the sky that in any way reminded me of a red rubber ball.
But it's a nice-enough song in spite of that and I can always make my friend Shelly Goldstein laugh by telling her, "Hey, you're not the only starfish in the sea."
I don't know what show this clip is from but I think it really is them lip-syncing. Back when I was doing variety shows, we usually had groups mime to their records or pre-recorded tracks and a very experienced TV director told me, "Very often, you'll find that the singers will do a good job but they'll be consistently a fraction of a second behind their voices on the track. That isn't a problem in itself because you can always slide the track a few frames and put their lip movements in perfect sync with the track. The trouble is when you have a drummer in the shot because most drummers are precisely on the beat so if you slide the track, you put the drummer out of sync. Usually, I have to split the difference but a better way is to just not get the drummer in the same shot as the singers if you can arrange it."
It doesn't look to me like they had that problem in this video but once the director told me that, I began to notice it on a lot of shows where singers were lip-syncing. They were in sync but the drummer wasn't — or vice-versa. And now that I've mentioned it here, you'll probably start noticing it.