Other people are better suited to most of your questions, but politics I can handle:
Assuming JFK wins in 1964 (likely, but not certain) against Goldwater or Rockefeller then we come to 1968. There are a lot of what ifs here, though.
If Viet Nam is a minor issue and JFK remains reasonably popular (no sex scandals, no revealing of his health problems—which is iffy) then the Democratic field is entirely different then OTL:
LBJ is the front runner, for sure, and RFK isn't going to run (he lives, of course, because of butterflies but still he would have been a better president then his brother IMO) and without Viet Nam neither McGovern nor McCarthy are major figures.
Humphrey on the other hand still is, but against LBJ it seems unlikely that he'd run or win.
So I imagine LBJ gets the nod.
If Viet Nam is a major issue then it's LBJ against McCarthy (maybe McGovern), probably, with Humphrey as a compromise candidate. Humphrey is in a much better position without being shackled by LBJ, LBJ probably does believe in JFK's Viet Nam war and so has that as a drag.
Nixon wouldn't run in 1964, but it seems quite likely he'd still gear up for 1968.
Rockefeller may do one of several things (depend if he ran and lost in 1964, or if he lost to Goldwater in the primaries): follow OTL plan of supporting George Romney, and then forced to enter the race himself because Romney wasn't that great; enter the race himself, give the nod to NY Mayor John Lindsay who didn't run in 1968 because he wasn't willing to go without Rockefeller's blessing.
Goldwater won't run again (regardless of whether he win or loses the '64 nomination he isn't going to win the general), and it's too early for Reagan.
Likely match-ups:
LBJ versus Nixon
It's LBJ's to lose on the following: Viet Nam, JFK's popularity, quality of campaign, and support from popular JFK, southern support, Chicago convention/riots.
Any of those go wrong in a big way and Nixon may win.
LBJ versus Rockefeller. Similar to above, except that Rockefeller is likely a better general candidate in the North, and weaker in the south—depending on VP to a moderate extent.
LBJ versus Lindsay. Lindsay is not as well known, but is otherwise similar to Rockefeller.
Humphrey is a weaker candidate then LBJ in all areas except Viet Nam. Nevertheless he has a decent shot against Nixon, assuming events go differently then OTL.
McCarthy and McGovern are noticeably weaker candidates in all areas except Viet Nam; they would likely lose any hypothetical '68 match-up against a Republican candidate.
The collection of what ifs expands massively if I have to go into 1972, but let me know.
Even if Nixon is in the White House the probability of Watergate is less likely as that depended on a lot of odd factors, and Nixon winning again in 1972 which is not as likely in a JFK lives universe.
As President:
Humphrey, McCarthy, McGovern, and Nixon (Rockefeller and Lindsay might, but would be more dependant on advisors) would all move to end Viet Nam if ongoing—I remain unsure about LBJ. Democrats would likely carry on with Civil Rights/Great Society type programs. Nixon would likely still go to China, and ignore domestic problems.
The possibility of a negative income tax replacing welfare/social security is likely under Nixon if he supports the replace part (as he didn't in OTL) and the Republicans have more Congressional strength. This is very much a good thing as regards economic growth and budgets.
Rockefeller/Lindsay are Teddy Roosevelt Republicans and would likely enact additional civil rights/some government spending. Lindsay, in particular, would pay attention to urban issues which the US badly needed at the time (not urban redevelopment like LBJ's Great Society, but local government money and new ideas. If butterflies make him read Jane Jacob's The Death and Life of Great American Cities the US may have some truly excellent cities going forward).
Rockefeller has no problem with government spending. Lacking Viet Nam he can also pay for it without major deficits like OTL, he may move to end Viet Nam if ongoing.
No one but Nixon would likely go to China, as most are less focused on foreign issues excepting ongoing Viet Nam.