Train transfers - how short is too short?

I wouldn't worry about the 7 minute connection in Levanto at all. There are several "Cinque Terre Express" trains per hour now running in the CT (which includes Levanto), so if you miss your train, take the next regional train (IC trains don't stop in Vernazza and wouldnt' be eligible for the same kind of fare anyway). Regional tickets can be used up to four hours after their initial time.

I had some other 7 minute connections last year in Italy that I made, but I experienced a lot of delays on the Italian Riviera trains, so I wouldn't be surprised if you miss yours. But again - don't worry about it. Just take the next one. Figure out which trains following are regional trains that stop in Vernazza (all trains heading to La Spezia go through Vernazza, but as I said, the IC trains won't stop there).

FYI, I found the Trenit app on my phone really helpful last year in checking real-time delays and platform info so I could be prepared while traveling for the next train.

Probably. 5 minutes is a typical allowance for a small station, where the other train may be on an adjacent platform and waiting for the passengers from your train, or even following it down the track, so it will be the next train into the same platform.
Big stations, like Milano Centrale, typical minimum allowances are 10 or 15 minutes.

From this photo, Levanto only looks to have 3 platforms: https://upload.wikimedia.org/wikipedia/commons/b/b5/Stazionelevanto.jpg

If you are still worried, most rail time websites allow you to set a minimum change time longer than the recommended, if you walk slower than average.

Thanks! Trenitalia app = downloaded!!

If you have to change platforms, then you will walk downstairs one level to the access tunnel and back up one flight.

We had a 7 minute connection in La Spezia, and I was a little concerned. When we arrived, it was so easy, we could have made a 3 minute connection.... I guess the only concern would be if you have other connections further down the line on a bigger train with a reserved ticket....

As Laura notes, you're likely to have to go down and then up a flight of stairs in Levanto. If you have a lot of luggage . . . . otherwise, you should be fine.