I wonder if it would help if I wrote a little program that plays short note sequences and then asks me to recreate the sequence (possibly modulo transposition).

Your ear (the ability to recognize intervals, scales, chords, chord progressions, rhythms etc) is one of (if not the) your biggest asset as a musician. And it's the only way to get the music out of your head, so cultivate it. Practice practice practice! Don't have to do more than a couple of minutes per day to see some results.

Ear training is something all creative musicians have to do. Many who are self-taught simply do it through picking up other people's tunes by ear (= transcribing), working it out on their instrument, starting with easy stuff and over time learning to "decipher" more difficult pieces. Dedicated ear training can really boost this process.