How to find similarity/distance matrix with mixed Continuous and Categorical data?
Similarity measures are subjective and so are they ways to combine them.
You should decide what is your subjective definition of similarity and then find a way to combine them that fit your definition.

In general, I like to reduce similarity problems into classification problems. Given the dataset of items you have, create a new dataset of item pairs. The concept should be whether the two items in a pair are similar. Each similarity measure you have is a feature of the pair. Note that now you are in the good old classification framework. You can evaluate the similarity measures by computing the mutual information/accuracy/your chosen metric given the concept. 

In your case, I would build the dataset this way. First choose pairs of items on which both your measures agree on. Choose pairs that are close by both the Euclidian distance and the cosine distance or pairs that are far by both measures. Note that since the similarity measures agree on these pair, combining them usually  leads to agreement too and not that important. However, you should use such pairs so they will stabilize your weighting function. Other than that, you can easily generate these pair, which is always an advantage. 

Next, you should choose pair on which your similarity function disagree. Choose some that are considered close by the first and far by the second and vice versa. Manually label the pairs in order to decide whether they are close or not.

Once you built the dataset, you can present your subjective definition of similarity as a set of pairs and distances. Now run any classifier (in your case a linear regression might fit what you look for) and use the result as the combined similarity measure.