I've developed a reputation for Federation political science, eh?
1) Assuming - and yes, I realize how unlikely this is, given that it's an alternate timeline - the same line of succession, who is supposed to be Federation President during the events of ST XI?
Weeeeeellll. Let's see. ST09 is set during six distinct time periods:
* 2233: Birth of James T. Kirk, attack on the
U.S.S. Kelvin
* c. 2243-ish: Kirk's and Spock's childhoods
* c. early 2250s: Spock's enlistment in Starfleet
* 2255: Kirk's enlistment in Starfleet;
U.S.S. Enterprise's construction at Riverside Shipywards
* 2258: Launch of the
U.S.S. Enterprise, destruction of Vulcan, battle against the
Narada
* 2387: Ignition of the Hobus star, destruction of Romulus and Remus, disappearance of Spock of Vulcan aboard the
Jellyfish and of the Romulan mining ship
Narada
We know from
Articles of the Federation that Madza Bral of Trill was elected Federation President -- the first one who wasn't from one of the five founding worlds -- about 150 years before 2380 and served two terms, so that puts her in the area of the
Kelvin attack. We know from
AotF's establishment of four-year terms, and from
A Time to Kill's establishing 2372 as an election year, that 2232 and 2228 would both have been election years (assuming that the terms of Federation Presidents were never altered).
So either way, I'd say there's a very strong probability that Madza Bral was President when the
Narada destroyed the
U.S.S. Kelvin (and the relationship between Vulcans and Romulans was subsequently discovered). Memory Beta conjectures that she was elected in 2232 and took office in early 2233 in order to accommodate a number of Presidents from RGP manuals.
Memory Beta indicates that
Star Trek Log Seven features Samuel Solomon Qasr, home planet unestablished, as Federation President in 2245 when the
U.S.S. Enterprise was launched in the Prime Timeline. The novel
Burning Dreams establishes that an unnamed female non-Human was President during the 2210s.
I'm afraid that the novels have not established any other Federation Presidents between Archer in the mid-2180s to 2190s and Wescott in at least 2264. As you'll see below, Wescott was probably elected in 2260. We know that Presidents don't have term limits, so Wescott could conceivably have been elected in 2256 to the 2257-2261 term, one year after Kirk enlisted in Starfleet and serving during the
Narada crisis. However, we have no info from the novels to base that on; it's pure speculation.
Memory Beta also includes information from the RPG manual
Star Fleet Intelligence Manual: Agent's Orientation Sourcebook alongside the info from the novels. These haven't been confirmed by the novels, but at least two Presidents featured in the novels -- McLaren and Gan Laikan of Alpha Centauri -- originated from other RPG manuals.
If we do expand our pallet to include the RPG Presidents and go by Memory Beta's conjectural dates for their terms, then the timeline is this:
2233-2241: Madza Bral of Trill
2241-2245: Paula Christenson of Earth
2245-2249: Samuel Solomon Qasr
2249-2253: Jacob Varis of Arcturus
2253-2257: Sanara Dadari of Earth
2257-2261: Claton Mintaine of Earth
Star Fleet Intelligence Manual: Agent's Orientation Sourcebook also has Sarboran of Vulcan serving from 2261-2265, but as I note below, that probably contradicts
Vanguard: Harbinger's implication that Kenneth Wescott served that term.
You are, however, completely correct in noting that the political ramifications of the destruction of the
Kelvin in Federation space at the hands of the
Narada may have been so profound as to cause entirely different results in every Federation election held thereafter.
We have no information from the novels about who is President in 2387, as they're still in 2381. The
Star Trek Online backstory has Nan Bacco continuing on until 2390, but STO is a completely different continuity from the novels.
2) What about TOS? Who was president then?
Well, we know from
Vanguard: Harbinger by David Mack that the President had been running for re-election in 2264. We know from the
Errand of Fury novels that Kenneth Wescott of Earth was Federation President in 2267. And we know from
Articles of the Federation that Presidents serve four-year terms, meaning that the President running for re-election in 2264 would take office in early 2265 and serve until early 2269. And we know from
Articles that Kenneth Wescott was succeeded by Lorne McLaren, who would have served at least one term lasting from 2269 to 2273.
Now, it's possible that the President running for re-election in 2264 is a separate President from Wescott, since Pennington's internal monologue doesn't name him. However, I think that it's unlikely that he would have been referred to as "the President," present tense, if he had not won re-election that year.
Going by the timeline in
Voyages of the Imagination to determine what year TOS episodes are set in, I would infer that the Federation President during "Where No Man Has Gone Before," Season One, Season Two, and the first half of Season Three up to "The Mark of Gideon" was Kenneth Wescott, with Lorne McLaren serving during the second half of Season Three (from about "The Lights of Zetar" or "The Cloud Minders"), and possibly during
The Motion Picture (depending on whether or not he served multiple terms and/or on how early in 2373 TMP took place).
3) About Hiram Roth (the ST IV president): I know that soon after the events of that film, Roth wins reelection in a landslide, but dies on the day the results come in. Which novel was that mentioned in? Does it say how Roth died?
That was established in
Articles of the Federation. It isn't established what he actually died of; President Bacco simply remarks that in the wake of the Probe incident from
Star Trek IV, he worked so hard to help Earth recover that he became ill and died the day his re-election was announced. The
Myriad Universes novel
The Chimes At Midnight feature President Roth dying of a massive heart attack during an attempt to flee Earth from a Klingon invasion; it's possible that the Roth of the Prime timeline suffered a similar heart attack.
Articles established that after Roth's death, the Federation had its first special presidential election, with Ra-ghoratreii of Efros winning the vote and serving three terms from 2289-2301.
1) According to Star Trek: Online, which conflicts with some novels, Bacco remains president until 2392, when she retires without endorsing a potential successor.
Star Trek Online doesn't just conflict with "some" novels, it conflicts with the entire continuity that the modern novels are based in. Amongst other things, the Borg Invasion of 2381 never happens in
Star Trek Online, which is a little bit like doing a story about 20th Century Europe where World War II never happens -- no matter how many similarities there are, your story is inevitably set in an alternate timeline because of its absence. There's also no Typhon Pact in the STO timeline.
So in terms of the novels, we have no idea who's President when the Hobus star goes wonky in 2387.
2) The president during The Original Series was Kenneth Wescott, who is reportedly based in no small part on President John F. Kennedy of the United States.
I always thought that President Wescott seemed very Kennedy-esque in
Errand of Fury.

When I read
Seeds of Rage, I imagined him being played by Bruce Greenwood, who played President Kennedy in the movie
Thirteen Days, with the Massachusetts accent and everything -- and this was years before Greenwood was signed to play Captain Pike in ST09.