How to interpret this logistic regression table?

As someone without much statistics training it can be very annoying to come across tables like this one (supposedly explaining the relationship between various individual level characteristics, such as age, class and perceptions of immigration rates, and the tendency to vote to leave the EU) in academic papers/books: There are a couple things bothering me here:

1. Can I compare the regression coefficients in order to rank the importance of different variables (e.g., can I say the perception that Brexit would reduce immigration is a more important predictor of Brexit voting than, say, believing immigrants are a burden on the welfare state, because the respective regression coefficients are 0.71 and 0.27, respectively, with both being statistically significant at the 0.01 level).
2. How can I describe the effect in layman's terms? In linear regression I think I'm right in saying that if the independent and dependent variables are measured in percentage terms then you can say a 1 per cent increase in the independent variable will lead to an X per cent increase in the dependent variable, but how does that work in logistic regression?
3. Why does there appear to be two different sets of regression coefficients (i.e., columns 1 and 3)?

Any help would be much appreciated.

• @mdewey provides precise answers to your questions. I assume this is the article: "Taking back control? Investigating the role of immigration in the 2016 vote for Brexit" by Goodwin and Milazzo (2017). Regarding your question 3, they clearly stated: "In the second model presented in Table 4, we include a variable that captures the intensity of each respondent’s anti-immigration sentiments across all three dimensions" (p.9). And they removed previous three variables. So, in this table, they reported the results of two different models. – T.E.G. Jan 19 '18 at 15:23

2. I am afraid your understanding of linear regression is awry. A unit change in the predictor leads to a $\hat\beta$ change in the outcome. The position for logistic regression is outlined elsewhere on this site in answers to the question linked below