I can answer most of those.
Your interpretations of items 1 and 7 are correct.
'Buck' can mean to writhe like a horse attempting to throw its rider, and also to oppose or resist in a more general sense. I take 'to buck agin you strong' [= strongly] as using the image of the former to convey the sense of the latter.
'Agin' = dialect form of 'against'.
'Small trumps' = low-ranking cards of the trump suit.
'Shuck' = the discarded husk of a vegetable, such as that which encloses an ear of corn (i.e. maize).
'Get skunked' = to lose a card game without winning even a single point/trick, etc.; this also fits in the context of the text you posted. (My Colorado-born wife also uses the term in that sense.)
On page 3989 of the Century Dictionary, I found several senses of the word 'nig'.
One of these is as a variant of the verb 'niggle'; one is as a verb meaning 'to be niggardly'; another is as a verb variant of the masonry term nidge, meaning 'to dress (a stone) with a pick or kevel' (kevel = a particular type of hammer); and what I originally suspected might be meant in the reference you found in the text, namely 'to clip (money)'. This sense of the verb is apparently related to the English dialect noun nig, meaning 'a small piece or chip', which is clearly relevant both to dressing stone and to clipping coins (in the days when some coins were made of gold or silver).
This explanation seemed plausible, given that it follows on immediately from a reference to clipping cards.
However, Robusto has provided an explanation in his comment that fits the card-playing context better still. I quote it as follows:
"...I believe nig is a shortening of the word renege, which means to trump a card instead of following suit when you still have a card of the led suit in your hand. For example, if someone led hearts and you had a heart you would be required to play a heart. If you trumped the trick instead (which is permissible only if you are out of the led suit), and were later caught at the subterfuge, you would be said to have reneged. Saying "don't nig" is another way of saying "don't cheat." "
On page 646 of the Century Dictionary, I found that one of the meanings of 'bower' is 'jack or knave', two of which are the highest-value cards in euchre.
So the father is telling his son that he must have the bowers (high-value cards) to back the low trumps; I assume the reference to the low trumps is therefore also in the context of playing euchre.
Finally, on page 2022 of the Century Dictionary, I discovered that 'cut-throat euchre' is a three-handed euchre game "in which one person plays against the other two together".
But exactly what the comment "it's regular 'cut throat' " is supposed to imply in the context of the warning against dishonesty that has just been given is still rather mysterious to me, beyond my inference that it too alludes to the game of euchre.