In this case, the use of 'of' and the use of the gerund are not directly linked. OxfordDictionaries.com gives the following as a definition of 'means' (emphasis my own):
(often means of/to do something) An action or system by which a result is achieved; a method
You could just as well use a regular noun in place of the gerund, as long as it was a noun that described some sort of action or event. For example, a 'means of payment' is a method by which payment is achieved, in the same way that a 'means of transporting animals' is a method by which transporting animals is achieved. 'Payment' is not a gerund, but it still works grammatically.
I think both of your interpretations of the first sentence are valid, although I think there is at least one other reasonable interpretation:
I have a number of means by which I can transport animals.
A 'means' is not necessarily a method that you actually use: it might just be a potential method that you could use.
Similarly, with the phrase 'moment of', you could just as well use a regular noun instead of a gerund. For example, 'the moment of realisation' is the event or point in time at which realisation occurs, in the same way that 'the moment of missing the opportunity' is the point in time at which missing the opportunity occurs. 'Realisation' is not a gerund, but it still works grammatically.
I think your interpretation of the second sentence is valid. Also, in my opinion, I think the interpretation is slightly better style than the original sentence, but either is valid.