Yes, you can. Specialization is a Wizard class feature that affects Wizard spellcasting only. It has no effect on spells from other classes. Complete Arcane, for example, explicitly suggests multiclassing with Sorcerer to get your missing spells.

Complete Arcane pg. 185

One way around at least part of this restriction is for a specialist wizard to take levels in sorcerer, using her sorcerer spellcasting ability to master the spells and magic items she cannot use as a wizard.

The FAQ (which is not a real rules source but may be good for convincing DMs) explicitly answered this question as well.

Specialization would be much clearer if they simply said that the spells were removed from your wizard spell list, and even effects that add spells to your spell list cannot add spells of your banned schools to your wizard spell list. That's what all the specialization rules sum up to, they just chose a long and complicated way to describe it.

However, note that Wizards, like all spellcasters, multiclass poorly. Being behind on spell levels is a very serious blow to your power. The addition of lower-level spells from another class is not nearly enough to justify this, from a purely-mechanical perspective. There are exceptions, but they involve dual-progression PrCs that can be entered with only one level lost, and that have their own class features (like Anima Mage from Tome of Magic or Ultimate Magus from Complete Mage).

What is open content is defined by what is literally present in the SRD itself. Ergo, yes, these are not open content currently.

Whether this is an oversight and other options were meant to be made open content, or whether the reference to the three monastic traditions is deliberate because they're leaving in references to non-open content, is to be seen.

But in general: content released in the SRD is what defines the content available under the OGL.