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- the grammatical gender will correspond to the gender of a noun
- if the noun happens to be masculine, then adjective is going to be in Genitive for animate and in Nominative for inanimate....

Isn't there only one noun in the whole sentence?

"Я нашла эту книгу" is the only correct variant. If such sentences raise questions, I would seriously recommend throwing away whatever grammarbook...

The verb usually means assign/set/define, and you use it in expressions like задавать диапазон значений "to assign a range of values" or задавать функцию преобразования "to assign a transform...

Yes, that's exactly what you need gender for. Adjectives agree in gender with a noun they describe.

1) Погодить is informal and, as far as I can tell, rarely used nowadays except for imperative "Погоди" (=wait a bit). Подождать is also used for that, and for everything else as well.

2) You...

Imperfective aspect doesn't imply the action wasn't completed, it just doesn't focus on this action as a certain point in time with a certain result. More as a repeated activity, or a prolonged...