Damn.  I just answered all your questions, then lost them.

Here it goes again.

First, I loved listening to the language in The New World, but had no idea
it was a reconstruction.  That's cool.  Are you actually trying to do
something like a full reconstruction/reimagination of a language?  If so,
I'm impressed.  The first thing you need is a the dictionary of
Proto-Algonkian.  You can find it on JSTOR I think, or at least a couple
papers on roots and their cognates in Cree, Fox, Anishinaabemowin, and
others.  Given what is known about the language already, using these roots
and comparing them to what you know and to what has been done in other
languages, you should be able to reconstruct the language at least to the
point where a fluent speaker would be able to guess at the meaning of all
the words you used, even new word that you make up for new concepts.  If
you were willing to fudge a bit and added a single small verb medial
meaning something abstract like 'technology' or 'electricity' you can
extend the vocabulary exponentially. :P

WARNING - the following is the product of an intense interest in the
subject and may not be interesting to everyone.

To explain obviation I'm going to start with the agent marking on verbs.

First, algonquian languages have 2 types of clauses, independent and
conjunct.  Independent clauses can stand independent, and conjunct cannot,
or choose not to.  They include all subordinate clauses, all clauses after
question words, and a lot of clauses that look like they could stand on
their own.  Here's an example using English, using the general rules from

John went to town (ind) to talk to a man about a dog (conj).  He saw the
man (conj).  The man gave him the dog. He then decided (conj) to come home

As you can see basically the independent clause can set up a new
paragraph, and every sentence in that paragraph or that train of thought
will be topically subordinate to the initial one.  Since independent and
conjunct sentences use almost entirely separate sets of morphemes, the
distinction is pretty important to learn.

Like you said, algonqian languages have an order of persons.  It goes 2nd,
1st, 3rd, 4th, with the 4th being the obviative (there is a 3rd person
inanimate used in the inverse set).  If I have a clause describing action
towards an object that is further down the order of persons, I use a
direct clause. ki-waapam-in, you see me. If the action is going the
opposite direction I use the exact same word, but insert an inverse
marker.  ki-waapam-iti-n.  You see (inverse) me.  There are different
inverse or direct markers for every set of persons (1st >2nd, 2nd >1st)
though a lot of them use suppletive portmanteau  morphemes that require a
lot of memorization and practice.

The 4th person (obviative) is essentially every 3rd person except for the
main 3rd person in a conversation.  This includes anything owned (I saw
John's dog), and all action of 3rd person upon other 3rd person/s.  (this
man saw that man).  There are completely different sets of determiners and
referrentials for obviative than for 3rd person, and it is used a lot.

The question is how to know who to make the original 3rd person. 
Generally it's something like this.  In the hypothetical train of thought
I created above, you would introduce your main 3rd person in the first
sentence.  Every new character after that would be in the obviative.  You
would also never have to restate John's name, and all 3rd person
determiners would refer to John.
In a lot of formal settings you can hear a long story where the main
character is introduced at the beginning, and never mentioned again, but
that's not how it works usually.

Most of the time, the different characters will be reintroduced every time
you have to use an independent sentence, or perhaps the other way around -
every time you have to introduce (think of it as pointing out someone) a
new character, or if you want to reintroduce a character, you have to use
the independent typle clause (And now he's gone (conj), where is he?
(conj) Here's over here!(ind)).

Much of the rest of these languages' grammar are also interconnected with
this concept - evidentials, determiners, pronouns, clause types, all can
be contingent on the framework built up in these sorts of large chunks of

The different languages have a fair bit of variation, but generally once
you understand one language well you can pick out the differences.  A
really good place to start is Clare Cook's phd thesis from UBC.  She
explains the whole system for Plains Cree, something that had more or less
just been guessed at previously.

And - since my conlang vocab is based largely on relexing the
proto-algonkian initials, medials and finals, using Cree or Michif to
gloss the new words actually was a lot easier than English!  I really
haven't described much of the grammar however as most of the grammar is
largely in my mind.  When I get it done I'll be posting it on the list.