You mean [in]dependent numeric variables, right? The dependent variable must be "infection" in your case, what is nominal ("yes" or "no") - otherwise the logistic regression would be off-topic.

I am fully with Martin and Luca regarding the outliers. However, if you have a lot of data and only very fwe "outliers" then their removal will not have severe evvect on the result - so you can leave them in. If you have only littel data, then having outliers is a double-problem. Unless the value is clinically/biologically/physically impossible (or highly implausible), it might be just the one telling you the important part of the story. Removing it might make your results "nicer", but also heavily biased. The only way out then is to report and discuss both results (with and without outliers removed).

You will not need a particular order of the levels of your nominal (independent) variable. The variable is nominal and remains nominal. To my knowledge there is no regression technique (model) that could handle ordinal predictors. As Luca said, you should just wisely chose the reference level, that's all. Because the effects are (typically, i.e. in "treatment-coding" of the contrasts) given as difference between all other levels to this reference level. You may also consider a different coding of the contrasts (e.g. "helmert" or "sum to zero").

You mean [in]dependent numeric variables, right? The dependent variable must be "infection" in your case, what is nominal ("yes" or "no") - otherwise the logistic regression would be off-topic.

I am fully with Martin and Luca regarding the outliers. However, if you have a lot of data and only very fwe "outliers" then their removal will not have severe evvect on the result - so you can leave them in. If you have only littel data, then having outliers is a double-problem. Unless the value is clinically/biologically/physically impossible (or highly implausible), it might be just the one telling you the important part of the story. Removing it might make your results "nicer", but also heavily biased. The only way out then is to report and discuss both results (with and without outliers removed).

You will not need a particular order of the levels of your nominal (independent) variable. The variable is nominal and remains nominal. To my knowledge there is no regression technique (model) that could handle ordinal predictors. As Luca said, you should just wisely chose the reference level, that's all. Because the effects are (typically, i.e. in "treatment-coding" of the contrasts) given as difference between all other levels to this reference level. You may also consider a different coding of the contrasts (e.g. "helmert" or "sum to zero").

I'm not sure how you would determine an outlier with a binary variable, particularly with only a categorical independent variable. In any case, make sure to provide a detailed account of all changes you made to the data in your final report/publication. 

Regarding your second question: choosing the reference category strongly depends on what you want to test. Luca's suggestion makes sense if you want to test whether any category differs from others, but if you want to test whether a specific category differs from a specific other category, you need to select one of these as reference.

In any case, when this is a categorical variable, the values do not matter, as long as your statistical software treats the values as categories (rather than as an interval variable). 

Choosing the reference category fully depends on your research question.

Regarding the recoding question. If your variable is "nominal" it has no inherent order of categories (e.g. colours) which is by definition the property of an "ordinal" variable (e.g. small, medium, large). So recoding changes nothing on your variable except that your categories have "new" names.

It seems you just recoded the categories by their mean outcome value. Now it seems as you would have an ordinal variable , but recoding based on the outcome makes no sense. Categories in your independent variable must have a definition by themselves and not based on the outcome that you want to associate with your independent variable.

Regarding the outliers I think Martin, Luca and Jochen said already everything. Especially consider Jochen's advice to report both, the analysis with and without the outliers and the observed changes in your estimates. This is a very simple form of sensitivity analysis.

I think there are really three questions here.  First, what to do about outliers.  But what is an outlier in this sense?  

So far we know that there are five categories (treatments perhaps) that lead to different outcomes (in this case, infection rates).  So assuming that the original categories were truly nominal, and assuming the dependent variable is also categorical (infected vs not infected), then I don't see how you can get outliers.  Or are you talking about some sort of infection rate being an outlier?

Second is the question of the recoding. This is completely irrelevant as you have nominals and what you call them makes no difference at all.  

Thirdly is the question of the reference category.  This has already been dealt with nicely by Rense.  You should think carefully about the hypotheses that you are testing and set up your reference to allow you to either test them directly or to make it simple to create contrasts that test them directly.

Your desire to recode the stimulus makes it sound to me as if you may be misunderstanding the logistic regression process.  From what you've said so far you have five conditions (whether they're inoculations, prophylactics, or treatments is irrelevant) and you are looking for a link between the condition and the infection rate, and more specifically whether these infection rates differ significantly from some baseline.

I wonder if there is not more information that you are leaving out.  It would help understand your question about outliers if we had a more complete picture of what you were doing.

Rense & Chris are absolutely right. If your dependent variable is binary, then you won't get any outliers. Or are you measuring actual infection rates? For continuous variables the advices of Jochen, Luca & Martin fully apply.

An easy way to assess the effects of an outlier is to run the model twice:  One time with the outlier and the second time without the outliers.  If there is little difference then I would include, if not I would exclude.