Colloquially, ~てしまう is contracted to ~ちゃう、and ~でしまう is contracted to ~じゃう. Both contractions conjugate like 五段 verbs. Example:

ごめん、宿題を忘れちゃった。Sorry, I accidentally forgot my homework.
家族の猫が昨日に死んじゃった。 Our cat died yesterday.
勉強した方がいい、私は日本語を話し方か忘れちゃうんで。It would be better if I studied, because I will forget how to speak Japanese.

Regardless of the correctness in my sentences, you can probably get the idea. :P

~ちゃいます・~ちゃいました is also possible, as a contraction of ~てしまいます・~てしまいました。

If you use noun + を + しまう, it means to finish using something and put it away:

もう着物をしまったよ。I already put the kimono away.

You can also use ~仕舞う with the meaning that a store or business closes for good.

for example:

不景気で売り上げは例年に比べって７５パーセント減ったので、店を仕舞うの仕方がない。

The second meaning first construction example just says ｢終わる｣ instead of 「終わっちゃう」.

How would one tell the differences between having finished something, and having finished something regretfully? ごはんを食べちゃった。(I regret eating the rice) 本を読んじゃう　(I read the book) Would it be based on the context?

It sould be 「て仕舞う」 With forced kanji it shows 「て仕舞」

The title. https://lh4.googleusercontent.com/-Ysc17j7SE3Y/UvBAiMcDy5I/AAAAAAAAAFU/bUJUhnFnqXg/w645-h543-no/screen.tiff

need more explanation....... :)

I am struck by the examples in which 終わる or 終える appear, combined with しまう, given that the meaning of all the verbs is "to finish, to finish doing". Isn't that too reiterative?

No, I understand the second meaning well. In it, as you say, しまう adds a nuance beyond its basic meaning. But in examples of the first meaning, as…

今週までにはそれを終えてしまいたかったのだが。

or

レポートを終わらせてしまった方が良いよ。

I get a bit of a feeling that しまう is really unnecessary. If しまう isn't going to add any special nuance to the sentence, and is only used for its meaning of "to finish, end something", combining it with 終わる or 終える doesn't seem necessary. It sounds like saying "finish and be done" or something like that (¿?)

I've been digging into this as well. My favorite Japanese grammar textbook gives 完成してしまった as an example in the completed category, so it does seem to be useable with verbs that also contain "finish" or "complete" as part of their meaning.