I started off with a toothpaste that didn't lather at all, and now it doesn't feel too weird. It takes some time getting used to, but it's better for your teeth in the long run!
Most people grow up rinsing, but essentially you're washing away all the fluoride
My dental hygienist would be giving you all disapproving looks and a disappointed sigh (and a printed out report with lots of orange and red on it to shame you into flossing)
Spoiler
and for the record
And I rinse the toothbrush and do a second brush rather than rinsing with water - best of both worlds
Toothpaste first and then under the tap, gotta say its never occurred to me to do it any other way. I also hope you are all turning the tap off after rinsing your toothbrush and not leaving it running while brushing your teeth.....