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I stumbled upon Nikolaych, which I assume is a short form of Nikolayevich.
Do all Russian patronymics have such forms?
Is there a simple rule to form them? Please give examples.
How are they used in real life? Only when addressing someone or also in written?
Do all Russian patronymics have such forms? No. For example, Ilyich and Petrovna are never reduced.
Is there a simple rule to form them? What you took for short forms are merely phonetic reductions. Therefore, they are formed by some kind of a natural process. When unstressed, elaborate suffixes such as -yevich, -ovich, -yevna, -ovna are reduced, the ultimate forms being -ich (i often being slightly iotified), -ych, -na, -vna: Andreyevich->Andreich, Ivanovich->Ivanych, Ivanovna->Ivanna, Nikolayevna->Nikolavna.
Reductions may affect the stem of the word: Mikhailovich->Mikhalych, Andreyevna->Andrevna. Name and patronymic of someone you're quite familiar with often form a tightly coupled compound which is pronounced as if it were a single word with stress on the name, heavily reduced: Mikhail Ivanovich->Mikhalyvanych, or even Mikhalvanch, Marya Ivanovna->Marivanna.
The sounds that fall out may not vanish entirely. In fact, when they do so, it is perceived as something somewhat coarse and rustic; however, between good friends that may be a sign of cordiality, reproach etc.
How are they used in real life? Clear articulation of unstressed -yevich, -ovich, -ovna or -yevna is done only to help someone with spelling. Otherwise they are reduced even in formal context. -ich and -inichna are always stressed and therefore pronounced quite clearly.
Only when addressing someone or also in written? Original forms of names and patronymics are almost always retained in writing (except for example when the writer wants to give a character's speech a kind of uneducated touch). There are also some special usages. For example, Marivanna is an archetypal teacher often appearing in Vovochka jokes.
If the father's name ends in -й (like Андрей, Сергей, Тимофей, Николай, etc.), the masculine form of the patronymic ends in an unstressed -евич [jɪvɪtʃ], e.g. Андреевич [ənˈdrejɪvɪtʃ]. In this case in the colloquial language this -евич [jɪvɪtʃ] can be shortened/reduced to -[й]ич [(j)ɪtʃ] with the final -й of the father's name dropping out or not, depending on the speaker, so
Андреевич [ənˈdrejɪvɪtʃ] > Андре[й]ич [ənˈdre(j)ɪtʃ]
Николаевич [nɪkəˈlɑjɪvɪtʃ] > Никола[й]ич [nɪkəˈlɑ(j)ɪtʃ]
If the father's name ends in another consonant (like Степан, Кирилл, Вадим, etc.), then the masculine form of the patronymic ends in an unstressed -ович [əvɪtʃ], e.g. Степанович [stɪ'pɑnəvɪtʃ]. In this case in the colloquial language this -ович [əvɪtʃ] can be shortened/reduced to -ыч [ɨtʃ], so
Степанович [stɪ'pɑnəvɪtʃ] > Степаныч [stɪ'pɑnɨtʃ]
Кириллович [kɪ'riləvɪtʃ] > Кирилыч [kɪ'rilɨtʃ]
This, however, does not apply to some names, e.g. although Олег ends in a consonant, you can hardly hear somebogy saying *Олегыч, since the sound combination гы is normally not found in Russian.
Also, Петрович has no reduced form, since Пётр is a monosyllabic name and the suffix -ович is stressed in it, so it cannot be reduced.
And some names have irregular colloquial patronymics mentioned by Schullz and user4419802 in the comments to the question:
Алексáндрович > Сáныч
Михáйлович > Михáлыч
Пáвлович > Пáлыч
