If I am reading this right you have a treatment (and theoretically a control) and then also a composite score from a survey. 

Are you trying to match assignment to treatment or control based on this composite score? If so that can be very time/resource intensive. If you are attempting to control for the covariate you could simply randomly assign participants to treatment and control and run an ANCOVA/regression unless there is a specific reason not to based on your research question.

With regards to your specific questions:

1. When possible I prefer to keep a continuous variable continuous as it is more powerful. Thus I would not perform a median split (again in general).  If you are using score to separate based on a criteria (e.g., depression) then there ought to be a clear delineation of your scores.  

2. Without knowing more about your specific example I don't know that I can answer this question. If you are looking at an assessment (e.g., depression inventory) and you want to ensure you have equal representation of "depressed" and "neurotypical" in each treatment then I can see utilizing a criteria to divide your participants - but that should be a part of the inventory's classification information.

3. I am not qualified to answer this question completely and thus will leave it to others.  

I don't understand your question. My understanding of the use of covariates is to remove nuisance variance that you cannot control experimentally. For example, if you're studying assertiveness in negotiations and you know that men and women differ in self-reported assertiveness (Cohen's d in one meta-analysis was > 1.0). You don't want to use only men or only women and you don't care to study this phenomenon (by making it a factor), so you use sex as a covariate.

I would generally urge researchers to avoid covariates unless they are necessary because any covariance between the covariate(s) and the outcome will decrease the effect size. Standard treatments of ANCOVA/MANCOVA simply tell practitioners to pick uncorrelated covariates, but the reality of that most variables are correlated in social sciences.

But I do have this answer about the 11 Likert items.  If they form a scale (i.e., they collectively indicate a latent variable) and you're interested in the latent variable measured by the scale, you're much better off using the scale score. The only advantage of dummy-coding each item would be if you were interested in the specific effects of each item.  For example, maybe the 11 items are each different symptoms of depression and you wanted to know the relationship of each symptom to the treatment.  Then it might make sense to do as you suggest. I think it might be just as easy to correlate each item with the outcome.

Above I said "it might make sense..." because if your items do collectively indicate a latent variable, you are almost always better off using the scale score. In order to find an effect for one item, you would have to find that the item measured something unrelated to the latent variable (i.e., that the scale was constructed in a shoddy manner), or that one or more items were biased. Biased items are actually fairly rare. A common example is about the frequency of crying on depression scales. Men and women of the same level of depression have different probabilities of admitting to crying (at least in Western cultures). Thus such items are biased. But a common finding of item bias research is that most items are not biased. I just helped a student examine the MCMI-2 for racial bias and she found just two items with both practical and statistical significance.

BTW, I see zero advantage to a median split unless it's just a convenience (after the analysis) to make graphing the relationship easier. I don't have the statistical equations handy, but it's well known that dichotomization reduces true-score variability of a measure without producing much benefit (e.g., Hunter's meta-analysis text treats this extensively).

Thank you very much for you answer. I can understand that the way I posed the question was not very clear. The reason is that I am only beginning to fully understand the issue hat hand.

I try to make a bit more clear what my intention is:

As Mr. Claxton suggested, my goal is to allocate subjects randomly to treatment and control, based on covariates. This is termed blocking, and is a form of conditional random treatment allocation. For example, a very simple binary covariate would be gender. The aim of this strategy, as ar as I know, is to enure that there is an approximately equal amount of males and females in the treatment, resp. control condition.

This strategy, as well as just measuring the covariate without blocking, allows me to investigate interaction effects of the covariate with the treatment. I.e. I can test whether males respond to the treatment differently than females.

In my case now, the covariate is an 11-item score. In my case, it is an 11-question questionnaire called “Hong’s Psychological Reactance Scale” (Hong & Faedda 1995). However, Mr. Mead did miscomprehend what I meant with the dummies. I am interested in the influence of the total score measured by the 11 items, of which each is a Likert type question with of one of 5 possible responses. I could now treat every of those 11 items as a single ordinal IV in regression, or I could dichotomize each item and treat each as a dummy IV. My usual strategy, however, is to dichotomize each item (coding responses with totally agree, and agree as 1, and all others as 0), and then adding up all eleven dummies to form the score. I think that this is a very heuristic approach and I am not sure if this is a common approach in psychology or other disciplines. I know that it is done in empirical economics.

Now my three questions concern the following: I create the score as explained above, and I want to form blocks by dividing subjects in two or three groups, e.g. >= median score and < median score.

Are there drawbacks? Are there ways to determine how to best divide the groups?

Thank you for clarifying. As best I can tell you should not dichotomize the subscores and instead sum their responses to create a single scale. Then you can run a moderation of group and scale (which one is the moderator will be dictated by theory, but my guess is the scale is the moderator) which will identify any interaction. You want to keep all the scale items intact because you have more detectable variance and keeping it continuous instead of binary also provides more power.

Edit: Also you shouldn't need to use the scale as a method of group assignment unless you have an extremely skewed distribution of scores in your population. That should all shake out through random (or pseudo random - i.e., alternating assignment) assignment.