It’s surprisingly doable. If you have a free weekend you can learn all of them. Even one day may be enough if you are dedicated and a quick learner. Of course, repetition is a must if you don’t want to forget them again, but the initial investment is really not that big.
This resource suggests mnemonics which are surprisingly effective. They teach hiragana first, but it’s not a hard requirement if you want to learn katakana exclusively.
Is the u pronounced? I recently learned katakana and it seems like the u variant is used to just get the consonant, i.e. if you pronounce words by leaving out the u they often match more closely to the English word than if you do pronounce the u (or to (ト), because tu doesn’t exist, it’s tsu). The two examples you gave match that thesis.
18+ What the Fuck Amazon?! (lemmings.world)
How do people understand each other? (startrek.website)