Yes, group 1 verbs always end in ~iru and ~eru. As you said, いる (要る）as in 'need' is not group 1, nor are the other words in your list. However, いる (居る) as in 'exist' is group 2.
The other exceptions can be found on this page: http://www.jgram.org/pages/viewOne.php?tagE=Group+1+Verb+Exceptions. There is quite a lot of them. But the examples you gave in your post are the more commonly used ones.
To repeat what the author says on that web page: There is a rule of thumb for judging whether a ~e/iru verb is group one or not. If there is more than one syllable before the ~ru, you look for the expression of the word with its kanji, e.g. 嘲る [あざける]. If the ~e/~i syllable is written in kana, it is group 2. Otherwise, it is group 1.
The author notes あぶらぎる and まじる are two exceptions to this rule. Also, if the word is only written in kana, you can't use the rule to determine which group it belongs to - e.g. bibi-ru and nome-ru. One syllable roots are also problematic when using this rule - e.g. he-ru (Group 1) and mi-ru (Group 2). ふせる could be either group, depending on how it's used.
Doesn't seem like there is a simpler rule. You just learn the exceptions from reading, speaking, listening and writing. People will (hopefully) correct your mistakes, so you can learn better.