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I tried to create a chart based on a few books, but it was largely guesswork.
Key: The first column of tables is for noun endings, the second is for pronouns, and the third is for adjective endings.
What corrections could I make to this table as far as conjugations?
Your question is in the list of candidates for closing with reason "primarily opinion-based". I agree that it really is a such; please try rephrasing it. Still, I'll try to answer this question as is, just in order to help.
The first thing that "strikes the eye" when seeing your page is: the cases are in "wrong" order. Typical order used in schools when learning cases is "Н-Р-Д-З-О-М-Кл" (Nom, Gen, Dat, Acc, Ins, Loc, Voc). IMHO, it would be easier for you to always list them in standard order, because: (a) most of books uses that order (by using standard order from the start you assure you won't need to re-adapt in future when you buy a new book); (b) using a fixed order makes learning easier by allowing some extra hacks (visual memory, use of mnemonics, etc).
You do a wrong thing by classifying nouns by grammatical gender. They really have genders — but if you want to learn declension (cases), you need other classification. (On the other hand, for adjectives it's good idea to make table by genders.) Nouns at the first level are classified into 4 types of declension:
"No-ending" practically mean that they end with consonant or soft sign.
Small letters are used as formatting for subsets that I suppose less important (ignore them in the beginning). They contain much smaller number of words and excluding them makes the scheme unambiguous (so that you can determine declension type for any noun just by its gender and nominative case ending).
"-я" and "-є" are put in parentheses, because they actually denote almost the same sounds as "-а" and "-е". (I. e. you can remember only rules for "-а" and "-е" endings, assuming "-я" and "-є" to be soft variations of "-а" and "-е".)
Nouns within the same declension type aren't necessarily declined absolutely identically, there are some subtypes. But at the first level nouns are to be divided into four declension types (not into three genders — when you learn declension). Somewhat more info can be found at the bold link above (if you can read Ukrainian).
The comprehensive rules can be found here (particularly here and here). It's a full list of all declension micro-types used in Ukrainian Wiktionary. Still, you probably don't need such a detailed level of specification — these micro-types are used for automatic declension by robots (not for humans, that are much smarter, e.g. can interpret "-а" and "-а́" and even "-я" as variations of the same, don't create separate type for every archaic exception, etc). Additionally, these micro-types use some unusual (maybe academic?) numbering style within their names (i.e. "2" in "імен uk 2*d f una" is not #2 from four "normal" declension types above).
puppy, kitten, piglet) probably are. I.e. IV-th declension is child-animal nouns and those of diminutives that are word-formed similarly to child-animal nouns.
dalekiy_obriy (that's one of the developers of Ukr. NLP in LT) or ask a separate question.
As far as any other language, Ukrainian has its own exceptions. For example, you can use endings "-а" and "-о" for nominative masculine nouns. A very similar and common example: "дядько" ('uncle') or "Микола" (Mykola, as the Ukrainan variant of the name Nicholas).
Neuter has also a possibility to be with the vowel: "дитя" ('a child'), "курча" ('a chicken') or "базікало" ('a chatterbox').
As well as feminine might have a consonant: "ніч" ('the night').
You can see this page on the Kyiv National Univ site for further reading.
