The bloody shoreline/House Rules
The rules for delaying an action in 5e are fairly simple (You can delay a single action until any point below your initiative. Plus something about a trigger that only half matters because it's contradicted.).
We're just going to even further simplify it even further. An initiative roll of 19 means that you can act at time 19 OR ANY LOWER TIME. You can't interrupt another turn to do so. Any effects that wear off on 'your next turn' still fade at the initiative you rolled (19 in our example), even if you delayed your last turn.
Rule of Cool
Doing something cool should result in advantage. In the case that your 'cool' thing involves trying to do damage with something other than your weapon, it'll still use your weapons stats - I don't want to make doing something cool less viable just because the numbers are lower.
In the case where the cool thing would infer some kind of penalty (ie. I smash my whiskey bottle in his face to blind him), I'd apply the blinding effect, but your standard weapon doesn't do that, so I'd run it like a normal improvised weapon.
So TLDR, it's situational, but for the most part it'll be just as effective, if not more effective from the advantage.
Receiving DM inspiration happens as usual (ie. Good RP, unique resolutions). Feel free to call me out for not awarding it. I hope the coins will remind me to, though. There is also an added thing we'll be doing in this game.
At the start of each session, each player writes a note of three things they want their characters to achieve. These can be long term, short term, a mix.
A completed goal awards inspiration, as do large steps made towards a long term goal. The standard rule of "If you have inspiration, you can't gain it" still applies, though!
These can be simple ("It's been a while since <character> got some, short term goal for this session is to get laid") or not so simple ("Arya has a list of names she repeats to herself every night" -- crossing a name off the list will award inspiration). The aim is to make characters have actual goals and things they want to get done.