(1) Attachment styles are set at an early age, but counter to what you said, being immediately available to children always does not guarantee secure attachment. The raging debate around the attachment parenting fad, and the glaring absence of any empirical support for its value, are a good indicator for this. Research into clinical interventions for correcting attachment disorders shows that attachment style is mutable, although hard to change. Interventions can, over time, morph people's attachment style, and in conjunction with that their narratives about their own childhood change too.
(2) Attachment doesn't come in 4 types. It has two axes (factors, in the sense of having being identified using factor analysis in questionnaires) -- "anxiety" a.k.a. "fearfulness", and "dismissal" a.k.a. "avoidance". People scoring low on both factors are labelled "secure"; but in reality there is a two-dimensional continuum with "secure" as one corner. These axes represent two dysfunctional ways of coping with "attachment crises": anxiously seeking reassurance, and pretending no crisis actually occurred.
(3) The notion of assessing attachment style in individual relationships is contentious; attachment style is part of personality and thought by many to be nearly immutable. However, attachment style only comes strongly into play when there's an attachment crisis, and what triggers our sense of such a crisis may be different with different people -- which might explain thedifferences that you observe in your personal experience. I'm not saying you're wrong about attachment being different in different relationships; I'm just saying that just because a popular website gives you different attachment scores for different relationships doesn't mean that there is any solid research to back this notion, which runs counter to the notion that attachment style is part of personality.
(4) (This is a side comment; and purely my own opinion) The "Strange Situation" is a laboratory induced attachment crisis. It has been argued that in a monogamously dominated society, polyamorous dating is a self inflicted barrage of attachment crises. This is why understanding adult attachment, and healing any residual attachment dysfunction, is critical for having happy poly relationships in a mononormative society. However, if you've been poly a long time, normal poly life -- like partners dating and falling in love with new people -- no longer induces attachment crises. My guess is that it is the very fact that it forces people to heal their attachment disorders that makes even an attempt at poly so useful for many people. I further suspect that in absence of the mononormative brainwash that creates an association between non-monogamy and partner loss, it will become much easier for people with all attachment styles to be poly.
(5) In direct opposition to what you said at the end of the program, avoidant individuals seem to do more poorly than anxious individuals in consensual non-monogamy (CNM; paper at http://spr.sagepub.com/content/32/2/222 ). The results of this paper are subtle and difficult to interpret, but they seem to show that while avoidant people find the notion of CNM appealing, they cope very badly with the attachment crises CNM relationships induce -- and thus end up no more likely to have a *successful* CNM relationship than people of any other attachment style.
I would highly recommend trying to contact Terri Conley, one of the senior authors of this paper, to help you better understand these topics and present them faithfully. She recently gave a fascinating TEDx talk ( https://youtu.be/185jZRFO83k ) which covers several human sexuality related topics you might enjoy.
(6) In the current attachment literature, jealousy and wanting one's partner to be exclusive are taken as indicators of secure attachment -- automatically pathologizing the attachment style of polyamorists. We are sorely in need of instruments (i.e. questionnaires) that assess attachment without implicitly enforcing mononormativity.