Learn Italian: Se fosse più preciso, avrebbe segnato. - If he were more precise, he would have scored.

This is a mixed conditional. We use it because the timeframe of the condition and the result are different. The condition (se fosse più preciso) is a general, ongoing truth: he is generally not a precise person. The result (avrebbe segnato), however, refers to a specific missed event in the past.

What would it mean if I said Se fosse stato più preciso instead?

That would be a standard Type 3 conditional (past condition, past result). Se fosse stato più preciso means "If he had been more precise" referring to that specific moment in the past. Using se fosse instead tells us that his lack of precision is a general, ongoing character trait.

Why doesn't the ending of segnato change to match the subject?

Because the helping verb for segnare is avere (conjugated here as avrebbe). When forming compound tenses with avere, the past participle stays in its default -o form and doesn't change to match the subject's gender or number.

Does the grammar change if I flip the order of the sentence?

No. You can say Avrebbe segnato se fosse più preciso with no change in meaning. The word se triggers the subjunctive mood here, so fosse stays attached to the se clause no matter which half of the sentence comes first.