Does drinking more than 3 glasses of milk a day double your risk of death?
The study was not designed to answer this question
The claim in your question title and in the headline of the Cleveland Clinic blog you linked ("Can Drinking Too Much Milk Make Your Bones More Brittle?") are both about causality. The study your question and this blog post are about is a cohort study, a prospective observational study that looks at baseline characteristics, or risk factors, and investigates their association with an outcome. In a traditional model (see Hulley's Designing Clinical Research) cohort studies identify potential risk factors. The impacts of modifying potential risk factors are then investigated in a different type of study, an interventional study, designed for causal inference. While a cohort study can use a variety of strategies to help enhance causal inference (see Chapter 9 in Hulley) there is always a risk of confounding by variables that aren't measured. There are a few specific problems with the study, but the study design itself is the main problem with the causal claim stated in your question. Though the study authors do advocate for a change in recommendations (which they shouldn't), the blog post you linked to does say:
while the study raises interesting questions, there is not strong enough evidence to warrant a restriction on milk
Does the study address potential confounders?
A confounding variable is a variable that is both associated with a causal variable and an outcome variable, but is not on the causal pathway. A classic example is the association between drinking coffee and developing pancreatic cancer. Drinking coffee is associated with developing pancreatic cancer, but this is only because people who smoke are more likely to drink coffee than people who don't smoke. Whether or not a smoker drinks coffee, they are more likely to develop pancreatic cancer than a nonsmoker.
The study addresses some confounders, but cannot dispense with the problem of potential confounding. You asked in an aside whether the study controlled for obesity. It did, by including BMI in the multivariate model, which you can see in the Statistical analysis subsection under methods. In general, the methods used in the statistical analysis were appropriate and can decrease the concern for confounding by the covariates in the multivariate model (see the second paragraph), but the problem remains: you cannot account for confounders not in the model. You can only do that by random assignment of a potential risk factor, which is an entirely different research design.
Other studies show the opposite association
Are there contradicting studies?
Yes. In addition to the meta-analysis mentioned in @Laurel's answer and the article you linked in the edit to your question, the largest and most comprehensive observational study of the impact of diary consumption on mortality, published online in the Lancet just a few days ago, shows the opposite effect. I would note that each of these studies have the same problem as the study making the opposite claim. They are not designed to tell you what will happen if you change your milk drinking habits, or, what is more relevant to the medical community, what will happen if a doctor or public health agency suggests you change your milk drinking habits.