Welcome to the world of NLP ^_^
All you need is a little basic knowledge and some tools.
There are already tools that will tell you if a word in a sentence is a noun, adjective or verb. They are called part-of-speech taggers. Typically, they take plaintext English as input, and output the word, its base form, and the part-of-speech. Here is the output of a popular UNIX part-of-speech tagger on the first sentence of your post:
$ echo "Without getting a degree in information retrieval, I'd like to know if there exists any algorithms for counting the frequency that words occur in a given body of text." | tree-tagger-english
# Word POS surface form
Without IN without
getting VVG get
a DT a
degree NN degree
in IN in
information NN information
retrieval NN retrieval
, , ,
I PP I
'd MD will
like VV like
to TO to
know VV know
if IN if
there EX there
exists VVZ exist
any DT any
algorithms NNS algorithm
for IN for
counting VVG count
the DT the
frequency NN frequency
that IN/that that
words NNS word
occur VVP occur
in IN in
a DT a
given VVN give
body NN body
of IN of
text NN text
. SENT .
As you can see, it identified "algorithms" as being the plural form (NNS) of "algorithm" and "exists" as being a conjugation (VBZ) of "exist." It also identified "a" and "the" as "determiners (DT)" -- another word for article. As you can see, the POS tagger also tokenized the punctuation.
To do everything but the last point on your list, you just need to run the text through a POS tagger, filter out the categories that don't interest you (determiners, pronouns, etc.) and count the frequencies of the base forms of the words.
Here are some popular POS taggers:
TreeTagger (binary only: Linux, Solaris, OS-X)
GENIA Tagger (C++: compile your self)
Stanford POS Tagger (Java)
To do the last thing on your list, you need more than just word-level information. An easy way to start is by counting sequences of words rather than just words themselves. These are called n-grams. A good place to start is UNIX for Poets. If you are willing to invest in a book on NLP, I would recommend Foundations of Statistical Natural Language Processing.