Former submarine officer here…
First off, with respect to the OP, the periscope does not come down when it is deployed; it comes up. In other words the whole periscope mast is extended up by hydraulics, so that the eyepiece and handles come up with the rest of the periscope mast out of the lower deck (floor).
Next, at least up through the improved Los Angeles-class (688-I) submarines, U.S. submarines still have a WW-II era periscope. That is, one of the two periscopes on the Los Angeles-class submarine is a Type 2 periscope whose design dates back to WWII. It was retained because it has a very small radar cross-section, and doesn’t have a many fancy accoutrements included, so there is less to break and go wrong.
The other periscope employed on a Los Angeles-class submarine is the high-tech Type 18 periscope, which has sophisticated electronics, built in communication and electronic surveillance equipment, night vision, higher magnification, etc.
More info here:
http://www.navy.mil/navydata/cno/n87/usw/issue_24/eyes2.htm
Also, periscopes are virtually always raised to their full height, which puts the eyepiece at eye level. You control the height of the scope above the surface by controlling the depth of the submarine. Both periscopes have the eyepiece directly attached to the rest of the periscope by fixed optics.
With respect to OP, the only reason to sling your arms over the periscope handles is for comfort. It’s like asking why someone holds a steering wheel with one hand at the 9-o-clock position vs. the driver’s ed method of two hands at 10 and 2 o’clock. There’s no real significance to the hand and arm position.
For a submarine either on the surface or at periscope depth, it is required that someone be continuously manning one or both scopes. As a junior officer, I spent hours at various times manning the periscope. After a few hours of “dancing with the one-eyed bitch” you hold the handles in just about every conceivable position that you could imagine.
BTW, there is very detailed scanning procedure that you use on a periscope that ensures that the entire horizon is scanned at different magnifications. It’s very tedious, but it is effective.
Finally, the latest class of submarine (Virginia-class) have moved to electronic periscopes (Photonics Masts) which is a whole other ballgame. I got out of the Navy just as these were being developed, but it’s my understanding that the periscope images can be sent to various displays in the control room, including large screen displays reminiscent of the main screen in Star Trek.