This is not some special construction, but is a simple combination of で and ね. ね is a sentence-final particle used to seek agreement or draw the listener's attention.
で can be either a case particle or the continuative form of the copula だ. As a case particle, で has many roles, so its translation can vary.
(That is) in Tokyo, you know.
I'll attempt to answer the question, but your confusion over the meaning is due to a general misconception you seem to have about how Kanji work.
You may have noticed that 償 has a Kun and an On reading, whereas 賠 only has an On reading. 償 can be part of a word, for example 償還【しょうかん】 where you would use the On reading. It can also be used by itself as the ...
I hardly think there is any difference there. However, as you mentioned, "でしょ" is only used in very casual situations, so it's best to avoid it outside of everyday conversation.
However, I personally think that "でしょ" tends to be used at the end of questions and "でしょう" at the end of affirmative sentences.
The key idea behind するのに is the idea of purpose. It has a different, more specific meaning than "することについては".
Don't think of するのに as a set phrase because it isn't - think of it as the purpose marker に, except the の is added to convert a preceding verb in する form back to a noun. As such, the する or の may not always be there.
Most of the time, (するの)に ...
Are they used by older people/in formal situations?
This depends on the word. Some are old enough and safe in business settings (e.g., メモる, トラブる, サボる). Some are rare and/or slangy. They are generally avoided in very formal legal documents, etc.
Are they really slang?
Generally yes, but many are widely used in day-to-day business settings, and there are ...