In regards to Trophy Rush, it's most likely under Games & More in the menu and not grouped with the Stadium games, similar to Coin Launcher.
Guessing from the Pic of the Day and the Video, the mode is probably default four minutes long. You break the mostly wooden blocks in order to fill a point or chain gauge, which when full would replace all or most of the blocks falling with coins to earn, and most likely an increased chance of customization items. Assuming the stone blocks are hard or impossible to destroy, they might take too much time and make you lose your chain, as well as interfere moving on the platform.
In regards to "Tin Pin Slammer" Mode, or at least what I'm calling it until it gets a name, I think it's either linked to a Streetpass feature for a small minigame, or grouped together with Trophy Rush as a mode primarily for getting coins and customization items, considering the coins that appear when the pins fly off the stage.
It's hard to get a lot out of the information we have though. It still could show up in a Pic of the Day or one of the last 3/4 Famitsu articles left.