Essentially your comparisons have the same meanings.
1. Being sentenced to death, he felt very sad.
2. Sentenced to death, he felt very sad.
However in #1, I feel the emphasis shifts from the actual sentence of the verdict to the judgment process of the verdict.
1. Being built in 1900, the bridge still looks new.
2. Built in 1900, the bridge still looks new.
I won't use 1#, simply because it may sound something like the bridge is still in the process of being built.
3. Being built, the shop has begun to start its business.
This sentence can also be understood in two ways: It has been built or It is being built. Would a half-built shop start its business? I must have seen such kind thing. Even if in this case, I think the sentence is still awful.