I think your concept of "kitchen table magic" is actually too restrictive. The main idea is to use cards you have access to while playing with friends, in any way possible. It's a very casual format where anything goes as long as the group agrees to it, and often includes house rules/restrictions to suit each individual table.
That being said, most people still follow the "constructed" deckbuilding ruleset, which is 60 card minimum, no more than 4 copies of any non-basic.
To answer your questions specifically:
Do you have to inform your casual opponent that you have cards from a commander deck in your deck or is it fine not to do so if you're playing "kitchen table magic"?
You don't "have to" do anything, but you probably should give your friends a heads up. Just talk with your play group about what they expect, especially for new cards.
Is it smart to include cards from commander decks to your main deck?
Kitchen table magic usually prioritizes hanging out with friends over building top-tier legacy decks. So it's smart if including them makes the game more enjoyable for you and your friends. It's a bad idea if including them will bring grief (some of these cards can be expensive, or poorly balanced for 1v1 settings).
What format do most people play generally? Modern or Standard? when playing "kitchen table magic"
Kitchen table magic is still usually constructed, so 1v1 with 60 card decks. So I guess technically, most people play legacy but won't have decks anywhere close to being competitive for tournament play.
There can be house rules, especially rules to ease deck construction (such as allowing proxies or free mulligans). Alternatively, there can be house bans on annoying cards that people don't like to play against (such as a "no 2-mana counterspell" restriction).
Even so, people often play various other multiplayer formats like free-for-all or two-headed giant.