The plus point with dropdown over checkbox is that their size remains same despite the selection(in case of non-multiple select). You can represent several options/data without occupying much space, which is not possible with radio/checkbox.
The drawback with dropdown is that most of the options are hidden from the user and they are visible only when the user opens the dropdown.
Coming to your questions:
So for what are type of data I should use dropdowns?
When there is a huge list of data and you have to show them without occupying much space. And, the data need not be present on the screen all the time.
Are checkboxes good for binary data like sex, availability of accommodation (yes/no)?
Use radio box instead of a checkbox for binary data
Also while I'm on topic of filters is slider to regulate price (like on 3rd filter on the image below) good UX?
Sliders are preferred only when the range is small ie the difference between the upper and lower limit is low. Otherwise, in case of large difference, it's difficult to select the exact value in the slider.
When designing such interfaces, you should also consider the development efforts and whether such designs are possible for the backend to replicate in the database. Lets discuss each filter:
I will suggest creating a range for the salary by dividing the numbers
into section ie 1 to 10, 11 to 20 etc. The reason is that it will be
difficult to set a maximum limit. Once, you have a higher maximum
limit the range slider will be difficult to use in the given small
area. So, a dropdown with range will be good.
Age can be a slider since you know the lower and upper limit and the
difference between them is not high
How will you handle transgender, any, all option? Also, when companies
post about jobs, they usually select "Any" or "All" in Sex field. You
should have handled that as well. The dropdown or checkboxes will be better here since you can offer the user to keep the field unselected while this will not be possible in radio boxes. Another point to note is that the radio, the checkbox will occupy more space compared to the dropdown.
This goes better with checkboxes since there are only two values.
Country & City:
I will prefer the design on the 2nd screen. A search in a long list is
a good feature. This will avoid the case where the user has to scroll
down the list to find their choice.
Since there can be either one selection for this option, it's better
to use radio box.
On a side note, try adding "Any" or "All" option in the necessary fields such as Accommodation, Worktime.