As far as I know, and I've researched this issue deeply in the past, there are no predictive modeling techniques (beside trees, XgBoost, etc.) that are designed to handle both types of input at the same time without simply transforming the type of the features.
Note that algorithms like Random Forest and XGBoost accept an input of mixed features, but they apply some logic to handle them during split of a node.
Make sure you understand the logic "under the hood" and that you're OK with whatever is happening in the black-box.
Yet, distance/kernel based models (e.g., K-NN, NN regression, support vector machines) can be used to handle mixed type feature space by defining a “special” distance function. Such that, for every feature, applies an appropriate distance metric (e.g., for a numeric feature we’ll calculate the Euclidean distance of 2 numbers while for a categorical feature we’ll simple calculate the overlap distance of 2 string values).
So, the distance/similarity between user $u_1$ and $u_2$ in feature $f_i$, as follows:
$d(u_1,u_2 )_{f_i}=(dis-categorical(u_1,u_2 )_{f_i} $ if feature $f_i$ is categorical,
$d(u_1,u_2 )_{f_i}=dis-numeric(u_1,u_2 )_{f_i} $ if feature $f_i$ is numerical. and 1 if feature $f_i$ is not defined in $u_1$ or $u_2$.
Some known distance function for categorical features:
What software are you using? Sounds like multinomial logit would be a good fit, if the categories are mutually exclusive and not ranked (for example, political party, university attended, etc.).
For Stata, that would be mlogit dependentvar independent1 independent2 [etc.]
If the relationship could go either way, I would definitely use caution in reporting results (like every stats teacher says, correlation does not equal causation!), and state why you first wanted to test the relationship this direction and not the other way.
Best Answer
Salary is typically treated as continuous even though there is a smallest unit by which it can be incremented: typically a cent. However this smallest increment is so small compared to the amounts that we are talking about that that is ignored. The bigger problem is that the distribution of salary is typically rather skewed, so you may want to apply a log transformation or a log link function. I tend to prefer the latter, see: http://blog.stata.com/2011/08/22
Days absent from work is a count, so you can try using count models (the simplest example would be Poisson regression). However, you probably don't have "days absent from work", but "days absent from work in the last week" or "days absent from work in the last year". That puts an upper bound on your days which strictly speaking you should take into account. In practice you can ignore that too, as the number of days absent is usually far from the number of work days in a year.
I know you only gave some examples, but there is a common theme: models are by definition simplifications of reality, so you need to think about potential problems and choose carefully which of these you choose to ignore. Then you can communicate your choice, and the reasons for it, to your audience, and the audience may or may not buy your arguments. Both are good outcomes: if they do then that is good for your ego, if they don't then you have learned something new.
Notice that the question what you want to do with the results plays a key role in these choices. So there can be no answer of the kind: "Model X is best".