[T]he problem with girls singing Anim Zmirot is that "my heart tells me" that this isn't right, because it harms "Kedushat Yisrael". Exactly as your heart tells you that there is no problem. There are no proofs for what is the "spirit of torah," if there was - it wouldn't be the spirit of Torah but halacha. There are many different opinions of Gedolei Yisrael in the past and in the present what is considered modest (צנוע), and when anyone who adds to the standard is only lessening Tzniut and when whoever adds will be blessed. You can cite examples to here and to here how to act, without ever proving the point. I have no problem that girls should sing Anim Zmirot as long as the person who enacts it is first class Gadol, who you can count on that his senses that this isn't against God's will (Spirit of torah)...I tried to keep the flowing style of writing in the translation, however I'm not sure I succeeded. The writer is probably best characterized as a very open Hardalnick, and as the paragraph shows he's very sensitive not to offend anyone. I liked both parts of his argument. The first part of his argument is merely a reflection of how he feels. He has a feeling girls should not be singing in a shul, but he does not ascribe it to halacha - just his gut feeling of what is proper. You can't argue with someone's gut.
One more point. I don't want to address the issue of modesty directly, but I'm sure you'll understand - there is a saying that only the left can go to war, and only the right can remove settlements. The left is not suspect of a messianic motivation for war, and the right is not suspect of a spiritual disconnect from Eretz Yisrael or ignoring security needs.
Lowering the standards separating boys and girls is not convincing in my opinion when it comes from certain people, of which a small minority have real problems in their daily manners (again I'm not thinking of anyone specific only generally and a small minority!). Some of them too easily stretch the boundary and even pass adopting a tolerant non thinking attitude to the relationship between the sexes, clothing and other similar concerns. Equally, and in the opposite direction, I am not excited when someone who always Machmirs in Kashrut publishes a new chumra. However, when a Chumra comes from someone who always tries to be lenient and despite that publishes a chumra - his opinion carries more weight...
The second argument is one I fully agree with - but not the conclusion. Like it or not, some people's motivation is slightly suspect. Many of us often have a feeling that some arguments with some strongly feminist members of the tribes are not really about spiritual fulfillment, but more about feminist ideology. It is very hard to know how to react to those situations. Should you dismiss their arguments only because YOU feel that their motivation is not "leshem Shamayim?" Can both motivations (feminist + Spiritual) exist at once? Do the two contradict? Is equality itself a spiritual motivation? Should you ever allow yourself to presume to know better than someone else what their "true" motivation is? I've never found any easy answers.