[Bootstrapping][1] is a resampling method to estimate the sampling distribution of your regression coefficients and therefore calculate the standard errors/confidence intervals of your regression coefficients. [This post][2] has a nice explanation. For a discussion of how many replications you need, see [this post.][3]

 1. The nonparametric bootstrap resamples repeatedly and randomly draws your
    observations **with replacement** (i.e. some observations are drawn only once, others multiple times and some never at all), then calculates the logistic
    regression and stores the coefficients. This is repeated $n$ times. So you'll end up with 10'000 different regression coefficients. These 10'000 coefficients can then be used to calculate their confidence itnervals. As a pseudo-random number generator is used, you could just set the seed to an arbitrary number to ensure that you have exactly the same results each time (see example below).
    To really have stable estimates, I would suggest more than 1000
    replications, maybe 10'000. You could run the bootstrap several
    times and see if the estimates change much whether you do 1000 or
    10'000 replications. In plain english: you should take replications until you reach convergence. If your bootstrap estimates vary between your estimates and the observed, single model, this could indicate that the observed model does not appropriately reflect the structure of your sample. The function `boot` in `R`, for example, puts out the "bias" which is the difference between the regression coefficients of your single model and the mean of the bootstrap samples.
 2. When performing the bootstrap, you are not interested in a single bootstrap sample, but in the distribution of statistics (e.g. regression coefficients) over the, say, 10'000 bootstrap samples.
 3. I'd say 10'000 is better than 1000. With modern Computers, this shouldn't pose a problem. In the example below, it took my PC around 45 seconds to draw 10'000 samples. This varies with your sample size of course. The bigger your sample size, the higher the number of iterations should be to ensure that every observation is taken into account.
 4. What do you mean "the results vary each time"? Recall that in every bootstrap step, the observations are newly drawn with replacement. Therefore, you're likely to end up with slightly different regression coefficients because your observations differ. But as I've said: you are not really interested in the result of a single bootstrap sample. When your number of replications is high enough, the bootstrap should yield very similar confidence intervals and point estimates every time.

Here is an example in `R`:

    library(boot)
    
    mydata <- read.csv("http://www.ats.ucla.edu/stat/data/binary.csv")
    
    head(mydata)
    
    mydata$rank <- factor(mydata$rank)
    
    my.mod <- glm(admit ~ gre + gpa + rank, data = mydata, family = "binomial")
    
    summary(my.mod)
    
    Coefficients:
                 Estimate Std. Error z value Pr(>|z|)    
    (Intercept) -3.989979   1.139951  -3.500 0.000465 ***
    gre          0.002264   0.001094   2.070 0.038465 *  
    gpa          0.804038   0.331819   2.423 0.015388 *  
    rank2       -0.675443   0.316490  -2.134 0.032829 *  
    rank3       -1.340204   0.345306  -3.881 0.000104 ***
    rank4       -1.551464   0.417832  -3.713 0.000205 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
    # Set up the non-parametric bootstrap
    
    logit.bootstrap <- function(data, indices) {
      
      d <- data[indices, ]
      fit <- glm(admit ~ gre + gpa + rank, data = d, family = "binomial")
      
      return(coef(fit))
    }
    
    set.seed(12345) # seed for the RNG to ensure that you get exactly the same results as here
    
    logit.boot <- boot(data=mydata, statistic=logit.bootstrap, R=10000) # 10'000 samples
    
    logit.boot
    
    Bootstrap Statistics :
            original        bias    std. error
    t1* -3.989979073 -7.217244e-02 1.165573039
    t2*  0.002264426  4.054579e-05 0.001146039
    t3*  0.804037549  1.440693e-02 0.354361032
    t4* -0.675442928 -8.845389e-03 0.329099277
    t5* -1.340203916 -1.977054e-02 0.359502576
    t6* -1.551463677 -4.720579e-02 0.444998099
    
    # Calculate confidence intervals (Bias corrected ="bca") for each coefficient
    
    boot.ci(logit.boot, type="bca", index=1) # intercept
    95%   (-6.292, -1.738 )  
    boot.ci(logit.boot, type="bca", index=2) # gre
    95%   ( 0.0000,  0.0045 ) 
    boot.ci(logit.boot, type="bca", index=3) # gpa
    95%   ( 0.1017,  1.4932 )
    boot.ci(logit.boot, type="bca", index=4) # rank2
    95%   (-1.3170, -0.0369 )
    boot.ci(logit.boot, type="bca", index=5) # rank3
    95%   (-2.040, -0.629 )
    boot.ci(logit.boot, type="bca", index=6) # rank4
    95%   (-2.425, -0.698 )


The bootstrap-ouput displays the original regression coefficients ("original") and their bias, which is the difference between the original coefficients and the bootstrapped ones. It also gives the standard errors. Note that they are bit larger than the original standard errors.

From the confidence intervals, the bias-corrected ("bca") are usually preferred. It gives the confidence intervals on the original scale. For confidence intervals for the odds ratios, just exponentiate the confidence limits.

   [1]: http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-bootstrapping.pdf
   [2]: http://stats.stackexchange.com/questions/26088/explaining-to-laypeople-why-bootstrapping-works
   [3]: http://stats.stackexchange.com/questions/33300/determining-sample-size-necessary-for-bootstrap-method-proposed-method