Not to be snarky, but don't do any data analysis without considering the context.
There are not many cases where this encoding make sense. Unless your data have a natural ordering and you feel/know that a value of 2 should be twice the effect of the category associated with a value of 1 (and so on), don't do this. In the link that I sent, there is this:
For example, when discussing failure modes of a piece of computer hardware, experts would be able to rank the severity of a type of failure on an integer scale. A minor failure might be scored as a “1” while a catastrophic failure mode could be given a score of “10” and so on.
A dummy variable (aka indicator variable) is usually binary 0/1 for a set of columns to replace the original non-numeric source data. An example using a predictor for the days of the week is here.
There are other ways to encode the data that aren't binary, but dummy variables are the most common method.
Not sure what you mean. Do you mean that the categories have a defined order? Can you give an example?