alot Alot (one word) is a common misspelling of a lot (two words). "[W]e all may write alot one day," says The American Heritage Guide to Contemporary Usage (2005), but for now "keep in mind that alot is still considered an error in print."
and etc. Because the abbreviation etc. (from the Latin et cetera) means "and so on," the word "and" is redundant. Anyway, rather avoid using etc. in your essays; it could give the impression that you simply can't think of anything else to add to a list.
anywheres Only Huckleberry Finn can get away with saying, "There warn't a sound anywheres," but on formal occasions drop the "s". If anywheres appears anywhere in your dictionary, it's probably labeled "nonstandard" or "dialectal."
could of Don't confuse this nonstandard form with the contraction could've. Could of (along with should of and would of) can and should be replaced by could have (and should have and would have).
hisself This alternative form of the reflexive pronoun himself is commonly heard in certain dialects, but in formal writing steer clear of hisself and theirself. In Middle and Early-Modern English these were regarded as good usage though.
furtherest The comparative form of far is farther or further. The superlative form is farthest or furthest. Nothing can be gained by combining the two forms.
irregardless This double negative (ir- at the beginning and -less at the end) may not deserve Bryan Garner's label of "semiliterate . . . barbarism," but he's probably right that in print it "should have been stamped out long ago" (Garner's Modern American Usage, 2009). Use regardless instead.
its' Its is a possessive pronoun (like his or her). It's is a contraction of it is or it has. Therefore, this leaves nothing for its' to do—so biff it.
let's us Let's us means "let us us." To avoid the repetition, write either "She lets us play in her yard" or "Let's play in her yard" or just use "Let us play".