In grade 5 theory, you have to convert between open and short score and vice-versa.

What happens in the short score when two notes are at the same pitch at the same time but are different durations? eg. Theory Of Music Examination Theory Paper Grade 5 1999 S, question 3 requires me to convert from open to short score. The first note in the soprano part is a minim G. The first note in the Alto part is a dotted minim G. Now, if they were both minims there would be a single minim head with two stems, but what about in this case? Are they shown side by side or is it implied from the duration of the other notes in the Sop part?

You could put them side by side but touching or use a tie.

I would show them side by side, with the soprano minim on the left, stem up, the alto minim on the right, stem down, and its dot in the space to the right of the note head, or the space below if the note is on the line. The remaining soprano note(s) will confirm the minim value.

A short score simply condenses all the parts into fewer staves. Provided all the notes from the instruments are played, I see no reason to show any particular note twice.

I think it's necessary to make a distinction between two things here. A short score needs to be absolutely clear about who did what in the original expanded score, still allowing individual parts to be extracted reliably. A reduction of a full score on the other hand may be made to allow a recognisable version of the original to be played on piano. In that case the intention is not to reproduce the original details exactly, but to produce an outline of the sort needed for rehearsing big choral works or instrumental concertos with piano.

An intermediate stage is the "piano conductor" score, giving an outline of the music playable on piano, usually with the addition of a lot of unplayable cues in little notes showing what else is going on, perhaps partly on the same staves but also often on a separate stave above.