Q1. The Wiktionary list of characters with the 冫 radical contains the following two characters: 冬, 冭. Where in these characters is the 冫? Are the two lines at the bottom supposed to be the ice radical?
Answer: You're right. That's true.
Q2. When I look at the entry for 永 in the Chinese dictionary app on my phone (Pleco), then it says that it's radical is 水 and that it has 冫 as a component. This makes no sense to me. If 水 was the radical, shouldn't it be written as 氵? And where is the 冫? I see that 永 differs from 水 by two lines at the top, but is this really supposed to be the ice "radical" 冫?
Answer: The radical of 永 is 水, but it doesn't have the radical 冫. The top of 永 has nothing to do with 冫.
Q3. In the Wiktionary entry for 冫 , it says that 冫 is a radical that has been simplified from 氷. What character is 氷? It doesn't seem to be a Chinese character. Is this some ancient form of 冰?
Answer: 氷 was the vulgar form of 冰 in ancient China. It's obsolete in China but adopted in Japan nowadays. As @AngelLeliel has pointed out, 冫 is not a radical simplified from 氷, but 仌.
This is a very good question, yet, as a native speaker I understand some of the "truth" are controversial. Anyway the facts will be listed here, but you don't have to agree with my subjective opinion :)
Fact 1. 次, 习 have the "ice radical"? No, it's not fine with experts in mainland China or people who use traditional Chinese.
In 现代汉语词典 (Modern Chinese Dictionary, which is considered as one of the most popular authoritative dictionary in mainland China), 次 is indexed under the radical 冫 and the radical 欠. However in 中華民國教育部重編國語辭典修訂本 (Chinese Dictionary Revised Edition, the authoritative one in Taiwan), it is indexed only under 欠.
The reason for Chinese Dictionary Revised Edition not listing 次 under radical 冫 is 次 has nothing to do with 冰. The ancient character forms of 次 are
from Shuowen, from bamboo scripts of Qin, from oracle scripts. It is clear that after later oracle scripts, 二 defines the main meaning of 次 as "the second". 说文段注 Duan's Annotation to Shuowen in Qing Dynasty said "當作从二，从欠。从二故爲次。" (次 should have radicals 二 and 欠. Having the radical 二 thus it's the second.) It is interesting that all major dictionaries haven't put 次 under the 二 radical. Personally I think that's because they all follow the convention of Shuowen.
The simplified character 习 is under the radicals 𠃌 (乙) and 冫 in the Modern Chinese Dictionary. However the corresponding traditional character 習 has the radical 羽 (feather of birds). It's clear:
from Shuowen, from oracle scripts, from bamboo scripts of Chu. However note that Shuowen's "从羽从白" (having the radicals 羽 and 白) theory is not true -- there're material evidence from oracle and bamboo scripts proving the bottom part is not 白 but 日. The character just illustrated "birds study flying on a sunny day", so it meant "study" (opinion from Guo Moruo's 卜辭通纂考釋).
Now you may feel dissatisfied for the way that dictionaries of simplified Chinese classify 次 and 习. However, if you forget the whole story above, it may be the only way you can find it in a dictionary for the simplified character forms. 二 in 次 or feather in 习 have been written as 冫 for a long history as it's more convenient (OK I think some experts may insist my referring to Taiwan's standard glyphs 國字標準字體 here but let's not go so far if you're not learning traditional Chinese). So if you don't care about the etymology, keep considering 冫 is the radical of 次 and 习 can be a self-consistent theory in your mind. I guess that's what the experts think when they compile the dictionaries for simplified Chinese.
Fact 2. A not-so-exact-but-easy theory "两点冰，三点水，四点火" -- two points denote "ice"; three points denote "water"; four points denote "fire".
Let's see the original character 仌 for 冰:
from bronze scripts, from Shuowen.
It illustrated the texture on ice, and became two horizontal strokes when coming to clerical scripts:
=> , => .
Thus when it came to regular scripts, the "two points" under 冬 and 冭 were written as two "down point", and the "two points" on the left of 冰 were written as 冫, possibly for the aesthetic reason.
Similarly, 氵 came from what 水 was written in bamboo scripts of Qin:
It is a rarely known fact that 氵 appeared in Qin state before the seal scripts became the standard in the early Qin Dynasty:
河注沙, the left component is 水 in seal scripts.
Thus, as 氵 and 水 stand for the same thing, water, they often shared the same entry of indices in ancient Chinese dictionaries like Kangxi Dictionary.
Similarly again, 灬 often stands for 火, fire. 灬 and 火 shared the same entry of indices in Kangxi Dictionary.
Fact 3. 永 has several versions of explanation, but none of them is related to "冫ice".
Shuowen: "永, long, illustrates the length of a river. Classic of Poetry says 'The length of the Jiang (river), cannot be navigated with a raft'"
Gao Hongjin's 中國字例 Examples of Chinese Characters: "Note: this character 永 is the original character of 泳 that denotes diving and swimming in the water. In the beginning its semantic element is people swimming in the water, the 人 and 彳 parts, so it's borrowed to mean swimming ... later it's borrowed to mean 長永 (permanent, with the same sound), and stuck to this meaning for a long time, so 水 was added making it 泳 to denote the original meaning swimming."
說文新證: "彳 denotes moving, 卜 illustrates a river. The whole character illustrates a running river, meaning the view of a long river. The shape of 卜 might be wrongly written as the shape of people 人, so curves are added to illustrate the shape of water."