As to the first two remarks: 'time spent reading a book' would indeed be categorical. In this case, your reasoning is fine. There are unequal and vague differences between the values which means the scale is not continuous. It is rather a set of ordered categories. Moreover, the number of different values is small (only 4 possible values), making it hard to say whether this variable could actually be modeled as continuous (whether its data generating mechanism is or not).
The same goes for the number of close friends.
For the third question/point see Regression for categorical independent variables and a continuous dependent one.
In short, for a general regression model the kind of independent variables (continuous or categorical) does not matter (i.e. it can be both), but note that the categorical data needs to be dummy-recoded for some software applications.
As to your last question, the type of dependent variable does matter. A linear regression will violate most, if not all, of its assumptions when used to study a binary dependent variable. Luckily for us, this is easily overcome by using 'generalized linear models'. A couple of examples: for binary outcomes this means you use logistic regression; for count data you use Poisson regression; and for categorical data with more than two levels you use multinomial regression when the dependent variable is not ordered, and ordinal regression when the dependent variable has more than two levels and is ordered. I feel the latter might apply to your situation.