Fridge Logic: Shinzon has a grudge against the Romulans for creating him, lumbering him with a genetic disease, and consigning him to a life of misery in the Reman mines. Therefore, he wants to utterly annihilate... er, Earth.
This was explained somewhat in the chunk of footage that was left on the editing room floor. Shinzon's lifespan was going to be pretty short even if he had succeeded in stealing Picard's blood, and he in fact killed most those responsible for his creation when he took out the Romulan Senate. His main goal was in fact to go down in history by destroying Earth, which would bring about the collapse of the Federation government and pave the way for a full-scale Romulan invasion. A bit of a hackneyed motive to be sure, but it makes much more sense than what we got.
His talk about the Remans being a race "bred for war" implies that he and they are basically a bunch of Blood Knights who want to start a war, this time on their own terms and not as slaves of the Romulan Empire. By extension they can be read as trying to show the Romulans up, taking out the Federation in one shot after centuries of similar Romulan plots failed. It's still pretty vague, though.
The Enterprise received six positronic signals, but only one of Data's components (his postrionic brain) would register such a signal. Each of B-4's parts also apparently has its own power source, if the "zombie hand from the sand" is any indicator.
Fridge Logic: Why isn't Worf the one to go on the suicide run on the Reman ship? He lived through the last big war, and his religion actually requires him to seek a glorious death. In the other movies Picard being the one to go in is due to circumstances or necessity, but here it makes no sense.
Because he was still fighting off the Reman boarding partym and in the novelization, a bad disruptor hit forced him into Sickbay; he was completely occupied in either case. Plus, he wasn't even on the bridge when Picard announced this decision, so he couldn't object even if he wanted to.
Fridge Horror: Admittedly, he doesn't want the promotion, but in what just universe is Captain Jean-Luc Picard, saviour of the Federation on more than one occasion, taking orders from a two-bit vice-admiral who was not even born when he was promoted to Captain and managed to screw up the one mission she had been officially given by Starfleet in seven years of Captaincy? (That is, search, locate, and apprehend a ship containing marquis and crew.) What does this say about the quality of Starfleet's Admirals???
That they reward people who unite two hostile crews, cross a considerable portion of a hostile, uncharted quadrant, making friendly contact with dozens of species, saved the galaxy from a species from another dimension or two and crippled the borg? And they don't punish people for getting grabbed by hyper advanced technology against their will. And hell, she DID bring them back like she was supposed to.
Some people, like SF Debris, have theorized that Janeway had gotten so exhausted after many years stranded in the Delta Quadrant that she couldn't bear to take another command, let alone immediately after finally getting home. Or that she was Kicked Upstairs to get rid of her.
Also, Starfleet may have figured that Picard is far more useful out there "on the field" than chained to a desk job. Picard has been the best at being the flag ship's captain since Kirk, and they want to profit from that as long as possible.
His Informed Ability as a tactician and command leader is just that; informed. However, despite being enslaved in the mines, the Remans do work with the Romulans on a daily basis, even if unwillingly... and above all else, Romulans are known as masters of deceit and lies. Bit-by-bit, the Remans could learn enough to emulate these strategies - so who's to say some other tactician led and won the mentioned Dominion War battles, was killed off, and the records were altered to make Shinzon take his place? With enough support/threats/bribery, they could also back up the illusion until it wasn't questioned anymore.
The rapidly-changing plan, first wanting to liberate his Reman brothers, then proving something to the RSE, and finally going to prove he's better than his clone. Well consider this: he was engineered as a clone, with temporal RNA sequencing designed to make him skip 30 years of his life, matching the original... which was never activated, causing a breakdown in his cells. And like the base of a pyramid, if the base started to become increasingly unstable, wouldn't it disturb everything above? Both physically AND mentally? Without the telepathic stablization from his Viceroy, one wonders if he'd still be sane at all.
In short, Shinzon is exactly what he appears to be: essentially a teenage boy trying to act like a man, trying to prove he's better than the original, AND desperately trying to prove his worth period before he expires. This with all the impulses (and desires) that come with that phase, summed up in the appropriate line "All I have are my personal feelings". And we see him steadily come apart as the film goes on.