I've heard that mycelium connect to each other when they are broken. Do they also connect with other types of mycelium to spread the network faster?
As you stated
...mycelium connect to each other when they are broken
This is true of mycelium in the same way that animals can heal a wound, an organism healing and reconnecting tissue.
The difference is of course that animals have a centralized physiology (centralized system for nutrient intake and managing of bodily functions), where as mycelial masses take in nutrients throughout their entire mass with no centralized governance.
Not only will a single mycelial mass reconnect itself when 'broken' if remaining in close enough proximity and with nutrients available to fuel the healing growth, but also a severed portion of the mycelium can grow independently into a 'clone' of the original organism when removed. Many species of fungus can also be cloned from the tissue of their fruit bodies (mushrooms) to grow new mycelial masses (mushrooms are in large part not much more than compressed mycelium).
Now, to the second part of your question:
Do they also connect with other types of mycelium to spread the network faster?
Let's try to address this incrementally.
Spore germination and growth: A typical single spore germinates into a homokaryotic mycelium, which cannot reproduce sexually; when two compatible homokaryotic mycelia join they form a dikaryotic mycelium. Spore grown homokaryotic mycelium are ONLY compatible if they are of the same species, this would be two mycelial masses of the same type connecting, however, two separate genetically distinct masses of dikaryotic mycelium of the same species (separate organisms) will continue to grow separately.
Additionally, just because a homokaryotic mycelial mass is of the same species as another that it meets, is not a guarantee that they will be compatible.
Competition between fungal species:
Fungi are incredibly competitive with other species of fungi.
A number of different scenarios may unfold when two fungal mycelia meet. If the two fungi are of the same species, they might mate. If they have already mated, then they often reach a kind of compromise, and a stand-off ensues, with neither advancing into the other's territory. If fungi of different species meet, battle commences with one eventually gaining the upper hand and advancing into the opponent's territory. Sometimes there will be stalemate if both are equally adept fighters.
Suggested further reading:
Mycelium Running: How Mushrooms Can Help Save the World Oct 1, 2005 by Paul Stamets
Radical Mycology, A Treatise On Seeing And Working With Fungi 2016 by Peter McCoy and Natassja Noell
Organic Mushroom Farming and Mycoremediation: Simple to Advanced and Experimental Techniques for Indoor and Outdoor Cultivation Aug 18, 2014 by Tradd Cotter