Which loss function should I use in my LSTM and why?
You'll want to use a logistic activation. This pushes each logit between 0 and 1, which represents the probability of that category. 
Then use categorical cross entropy. This will not make your model a single class classifier since you are using the logistic activation rather than the softmax activation. 
As a rule of thumb: 

logistic activation pushes values between 0 and 1
softmax pushes values between 0 and 1 AND makes them a valid probability distribution (sum to 1)
cross entropy calculates the difference between distributions of any type.
I try to understand Keras and LSTMs step by step. Right now I build an LSTM there the input is a sentence and the output is an array of five values which can each be 0 or 1.
Example:
Input sentence: 'I hate cookies'
Output example: [0,0,1,0,1]
For this, I am using keras library.
Now I am not sure which loss function I should use. Right now I just know two predefined loss functions a little bit better and both seem not to be good for my example:
Binary cross entropy: Good if I have a output of just 0 or 1
Categorical cross entropy: Good if I have an output of an array with one 1 and all other values being 0.
Both functions would not make any sense for my example. What would you use and why?
Edit
Another Question: Which Activation function would you use in Keras?