This is a very interesting discussion and I applaud you michaela for your questions. Always good to question to understand the teachings and practices better!
A few things to note:
I'd like to stress again that while Dorje Shugden arose principally to protect the teachings of the Gelug lineage, he is of course an enlightened being and would therefore help anyone who sincerely requests his help - Theravadans, hindus, christians and all included! Do remember that historically, Dorje Shugden was also widely propitiated by the Sakya school of Tibetan Buddhism too.
Secondly, we must be clear that it is NOT because the lamas, the Buddhas or protectors will be angered by us if we take teachings from other sects. This is a widely spread misunderstanding that has unfortunately developed into a wrong view that Dorje Shugden encourages sectarianism. This is absolutely not true. This practice of adhering to the teachings within one's lineage isn't just some question of loyalty. It is common sense. A gross comparison would be this - you don't go to Harvard AND Cambridge AND Oxford AND Yale universities at the same time. You stick with your course of education in Harvard and only when you have a firm foundation or are advised by your teachers would you begin to attend teachings from other schools. This is something that is practice across ALL lineages of Buddhism. A Theravadan would most likely stick with the teachings of the Theravadan traditions; a zen monk would engage principally in the teachings of zen etc.
This article might also help your understanding of this:
Does Dorje Shugden encourage sectarian practices?http://www.dorjeshugden.com/all-articles/the-controversy/does-dorje-shugden-encourage-sectarian-practices/Now, understanding that, we must understand that although it is extremely powerful and efficacious, our main practices should not be focused on Dharma Protector practices. Ultimately, we should aim to rely upon the meditations and practices of a Yidam (meditation deity), lojong, the Lamrim etc. for those are the direct paths to enlightenment. The Dharma Protector practices are commonly regarded as auxilliary / supportive practices that help to remove obstacles along those direct paths and to provide the conducive conditions for us to traverse it more quickly.
So it would therefore follow that if we are receiving an initiation into a Protector practice, we should be quite advanced in other principal practices and commitments / vows within that particular school / lineage and would therefore require a "stronger" protector practice to support us. In this case, it would make sense that we are already firm within the lineage itself and consider ourselves to be followers / practitioners of the Gelug lineage, in this case. (This would apply equally for practitioners in the Sakya tradition who might also receive Dorje Shugden's initiation).
Having said that, anyone can still make a connection to Dorje Shugden by making offerings, and reciting simple prayers that can be recited without initiation. When we do this, we remove immediate obstacles to our spiritual journey and at the same time, create the causes to be able to find qualified teachers, centers, practices and a lineage / school where we can eventually, ultimately engage in deeper, stronger practices to lead us to enlightenment and freedom. Even if we feel we are "not ready" to commit to a lineage or school yet, Dorje Shugden will still help us to remove our immediate worldly obstacles and help us to find a way to a school and teacher that would be most karmically suited and beneficial to us and our spiritual journey. A possible non-Gelug scenario: perhaps if we were strong Nyingma practitioners in a previous life and our affinity and karma is within that lineage, then praying to Dorje Shugden in this lifetime may lead us to finding a Nyingma Guru, center, and teachings for us to reconnect to the practices and lineage we pursued in our previous life. Dorje Shugden is not "greedy" for people to join just the Gelug school haha. As an enlightened being, his practice will lead us to what is most beneficial for us.
And anyway, once we're all enlightened, what difference does it really make if we got there by being Nyingma, Theravadans, zen or Gelug?