Q:

Can an igloo be a valid succah?

Follow up to this answer. The end of item 1 encouraged me to ask a question regarding using ice as the succah walls. Assume that scrub brush is used as schach.
Some additional parameters regarding ice:

If it is likely to melt, I assume it would be a problem using this to begin with, as the walls are not steady. What if you live in the arctic where there is a low likelihood for melting during Succot?
What if there is a high likelihood of heavy snow on a nearly daily basis. Enough snow to cover the schach such that no sunlight will enter the succah?
While the ice walls are not likely to melt, snow tends to sublimate, actually, Thus, the thickness of the walls may change, though still its likely that the walls will be thick enough to support the schach. Is there any requirement that your succah walls maintan the same thickness throughout succot or can it erode?
You can assume in this case that the igloo is not circular and has at least 3 walls. I don't think a circular succah is valid.

A:

Regarding #1:

If it is likely to melt, I assume it would be a problem using this to begin with, as the walls are not steady.

Not sure what you mean here. If the walls don't sway in the wind, then they are halachically steady. (And see #3)
Regarding #2:

What if there is a high likelihood of heavy snow on a nearly daily basis. Enough snow to cover the schach such that no sunlight will enter the succah?

The Be'er Heitev says in סימן תרכו - העושה סכה תחת האילן או תחת הגג, when the Shulchan Aruch says:

א:  אֵין לַעֲשׂוֹת סֻכָּה תַּחַת (א) בַּיִת אוֹ אִילָן‏
באר היטב  (א) בית.  דבעי שתהא הסוכה תחת אויר השמים. מקומות קרים שבחג יורד שלג הרבה על סכך הסוכה ומתעבה ונעשה כעין תקרה כשרה. גינת ורדי' חא''ח כלל ד' סי' ז': ‏

In places where it snows and the snow becomes as thick as a roof, the Sukka is still kosher.
Regarding #3:

Is there any requirement that your succah walls maintain the same thickness throughout succot or can it erode?

The Remo in סימן תרל - דיני דפנות הסכה says that the walls may shrink during Sukkoth, on condition that by the end of Sukkoth they are still kosher walls (i.e. 10 tefachim high, not swaying in the wind, no 3 tefachim spaces, etc.).

א: כָּל הַדְּבָרִים כְּשֵׁרִים  לִדְפָנוֹת וְאֵינוֹ צָרִיךְ לְהַעֲמִידָם דֶּרֶךְ גְּדִילָתָן (טוּר), וַאֲפִלּוּ חַמָּתָהּ מְרֻבָּה מִצִּלָּתָהּ מֵחֲמַת הַדְּפָנוֹת, כְּשֵׁרָה. הגה: מִכָּל מָקוֹם לֹא יַעֲשֶׂה הַדְּפָנוֹת בְּדָּבָר שֶׁרֵיחוֹ רַע (רַ''ן), אוֹ דָּבָר שֶׁמִּתְיַבֵּשׁ תּוֹךְ ז' וְלֹא יְהֵא בּוֹ שִׁעוּר מְחִצָּה (דִּבְרֵי עַצְמוֹ).‏

Regarding #4:

I don't think a circular succah is valid.

The Shulchan Aruch in סימן תרלד says that a circular Sukka is kosher, if it is big enough to hold a 7x7 tefachim square.

ב: אִם הִיא עֲגֻלָּה, צָרִיךְ שֶׁיְּהֵא בָּהּ כְּדֵי (ב) לְרַבֵּעַ שִׁבְעָה עַל שִׁבְעָה.‏

The Be'er Heitev explains that it would need a circumference of 29.4 Tefachim. Presumably on the inside of the igloo, in this case.

באר היטב  (ב) לרבע. וחוט סובב כ''ט טפחים וב' חומשים הוי שבע' על שבע'. ב''י ועיין ט''ז ומגן אברהם: ‏

Summary:
An igloo - with Kosher s'chach  - would be kosher, if it could survive the duration of Sukkoth.

