It's a great question! Your impression is certainly the way the Stargate was supposed to work, but, the writing team needed a variety of different elements to have some consistency throughout the Stargate series', and because of this, the Stargate itself ends up being nearly identical in its' transportation function as the transporters from StarTrek. The obvious exception being, well, obvious.
An object enters the Stargate, is then disassembled into it's most basic parts, stored as a series of data and transmitted via worm-hole to the receiving gate where it is then reassembled. That is not how the Stargate was originally presented, but it is the presentation that eventually became canonical.
The Stargate was supposed to "shunt" an object through a worm-hole to the receiving gate without disassembly and reassembly. If that were the case - if the creators had stuck with that idea - then you would be correct, it would eliminate the moral quandaries presented by the transporters from Star Trek. Since they didn't, it's still present in Stargate. In fact, the notion of mass copying (granted of a wave form rather than a physical object), is a lynch pin in one of the two-part episodes very late in the original series run.
TL;DR to the edit: It's uncertain.
It is unclear from footage in any of the Stargate series whether or not the experience of the individual is that of simply "stepping through", mostly because it is inconsistently displayed. On the one hand, you have people stepping in on one side and stepping out on the other, implying that they "feel" like they're just walking through. On the other hand (and used significantly less often because of budgetary concerns), you have an individual stepping in, the "worm-hole effect" graphics display, and then they step out on the other side, implying that the individual does experience some sort of event during the transition.