Unless you are going to be a Farsi teacher, stick with Sarah's answer's last paragraph. Both above answers are meticulously correct but the fact is that most if not all Persian speakers use the words you have asked about, without any knowledge or regard for their inner construction. They simply take them as independent words. After all "che" and "ak" are not that alive in contemporary Persian. that means we use them only in the words that have been coined long before but we do not make up new words with them. all in all whatever word containing "ak" and "che" is just another new vocabulary you should learn.

what the book said about animate objects and inanimate objects is right, but you should keep it in your mind that in Persian it depends on the object and word actually, there are many words you can't use neither che nor ak but in Persian you always can say ketab-e-koochak as an adjective or setare-ye-bozorg , in informal speaking mostly for children it is also possible to say: object+koochooloo ,so as a matter of fact you should learn them in practice although, generally using adjective form is absolutely right, for example: for shoe we can't use any of mentioned suffixes, as an instance kafshche and kafshak are both wrong in Persian, for augmentatives your're right we don't have any suffix but adjectives. more examples if can help u: dorkhtarak(for girl), pesarak(for boy) dokhtar kochooloo or pesar koochooloo(as mentioned above) daftarche(for notebook), ketabche(for book), anyway you'de better suppose them all as an separate word cause sometime you may see a word with che or ak at the end but it doesn't mean that it is composed with suffix like, dibacheh(means introduction) or kharak(which has some meanings like sawhorse but it doesn't mean khar+ak at all!"as you know khar means donkey in Persian")

1- If I want to make a diminutive of "shoe"/"star"/"book", can I use whichever I want to, or is every word formed with a specific suffix? For most nouns, we use an adjective to indicate the smallness. Small shoe, star, bag, house, ... "Che" is used in less than 20 nouns which about 10 you may need to know, the rest are unusual. derakhtche, bazarche, ketabche, daftarche, daryache?, ghaliche, bilche, baghche. "ak" is a bit complicated. Sometimes it is used to belittle or disrespect someone: mardak, dokhtarak, pesarak, zanak. Sometimes it shows compassion: teflak, dokhtarak, pesarak (depends on context) Rarely indicates the size: otaghak. 2- And what is the contrast in animacy, living-nonliving or mobile-immobile? (For example, plants are alive, but immobile so can "-če" be used with them or not?) I think mobile-immobile (we use "che" for derakhtche). 3-The book doesn't mention augmentatives. I suppose that means you don't have true augmentatives? Is there perhaps some other way you express augmentation other than syntagmatic? We have it, but not really classified in grammar and not very common to use. It is used only for some words. a) adding "shah" to some nouns: shah loole, shah rah, shah kar. b) adding "khar" to some nouns: khar pool (colloquially: rich), khar khoon (colloquially: bookworm), There are also some other prefixes that are not really recognized even by native Persian speakers. I advice you just learn and suppose these words as a separate word. They are not too many and don't worth to spend time to learn the grammar. e.g. "abar ghodrat", "Nare ghool", "abar mard".