Why would Reliant allow both her captain and first-officer to beam to the surface of Ceti Alpha V?
Why not? The film is set in 2285, not the Next Gen
era of apparently stricter and more formal shipboard rules and traditions for certain senior officers participating in landing parties and away teams. This isn't that many years after TOS, in which it seemed every Starfleet captain, his grandma and pet goldfish beamed down with their landing parties and took the same risks if not greater than their junior officers and security personnel. To me the behavior of Terrell and Chekov in beaming down together to Ceti Alpha V makes perfect sense in context of 23rd century Starfleet behavior that we became accustomed to in the Original Series and Animated Series.
And as per Spock Prime thinking that Pine Kirk looks just like Shatner Kirk, of course he would. Spock recognized nuKirk instantly when he turned around in the cave on Delta Vega and laid eyes on the man he had just rescued from being turned into CGI monster chow. Chris Pine's James T. Kirk is supposed to look just like William Shatner's and vice versa. I thought this was naturally assumed and the producer's intent all along. It's certainly what I've thought since I first saw the 2009 film so it's not anything big nor open for debate.
The only reason we
don't think Chris Pine Kirk looks just like young, TOS-era Shatner Kirk is because they're different actors. Our perceptions aren't what's important, though....the characters are the same. The face Spock Prime saw in that cave was the face he hadn't seen in the flesh in almost a century. The one he served many years with and the face of the man who was the greatest friend he'd ever known.