A Normal cConversation

In titles, it is generally normal to capitalize the first letter of any word that isn't too grammatical. That is to say, don't capitalize words like "the", "of", "in", or "a", unless they are the first word. Otherwise, capitalize, and in this case, saying "Normal Conversation", without any article isn't wrong, it sounds unnatural.

I had a barbecue party in the yard of my wife's parents' houseat my in-laws' yard in Shizuoka on the last weekend.

The fact that the yard is of the house of your parents-in-law (i.e. your spouse's parents), can not only be assumed, but is also preferably said without using "of". Generally speaking, use of "of" like that, where you could use normal possessive markings is a little archaic, and based on your story, I don't think that the tone fits.

We had a hot pot besidesalong with the barbecue.

I was looking for an electrical outlet for athe plug of an induction cooker.

Again, based on the fact that it sounds like you're writing a kind of jokey/ quippy dialogue, you should drop those things which don't need to be said. So, in this case, there isn't any confusion about the kind of outlet you need, plus I don't think I've ever heard anyone casually call one that anyway

Also, in general, use "the" if there is only one of something, and "a" if it is one of many, or not any particular one or type. In this case, unless this is a highly complex hot plate, it only would have one plug, therefore it is the plug.

A Japanese outlet has two sockets basicallybasically has two holes.¶
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Or you could say¶
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Japanese plugs generally have two prongs.

In trying to write a correction, I've realized that I'm having a hard time, not only because it is just a basic clarification, but also because in the US, most plugs have two prongs, but a lot of outlets have three holes, so it seems weird to talk about the outlet, when the punchline is about the plug.

My wife: How abWhy don't yout pluging it into in your nostrils? ¶
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You might want to use "nose" instead of "nostrils", just because you would have a symmetry of the plug (singular) and the nose (singular) like how you would plug a plug into a socket.

"why don't you..." Is a much more common way of quippily saying that someone should do something, especially when they don't mean it literally. "How about you..." Sounds a little too aggressive, since it's usually used saying things like "how about you f*ck off". "Why don't you shove it up your *ss is also perfectly natural, but it has more tonal variance in general, and really "how about you" would probably fit better there.

Me: If the cookerat would turns on with themthe cooker, I willould be happy to, anytime, anywhere.¶
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Or¶
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... I would do so, anytime, anywhere.

In the conditional statement, the first thing you talk about generally gets focus or emphasis, and in this case it sounds more natural to stress the action of plugging something into your nose, and not the cooker turning on. Also, while not wrong, using "to do so" sounds a little stiff. It certainly happens, but usually in much more formal tones and contexts.

The party was held without further incidents.

While this sentence makes sense, in the context of this, I think there might be a more finally consistent way to end it.

Last weekend I had a barbecue party in the yard of my wife's parents' houseyard in Shizuoka on the last weekend.

We had a hot pot besidesIn addition to barbecque, we also had hot pot.

I was looking for an electrical outlet for ato plug of anin the induction cooker.

Japanese outlets typically hasve two sockets basicallyholes.

My wife: How about plugging it in your nostrils?

Me: If the cookermy nostrils would turns on with themthe cooker, I willould be happy to do so, anytime, anywhere.!

I had a barbecue party in the yard of my wife's parents' house in Shizuoka on the last weekend.

We had a hot pot besidesin addition to barbecue.

I was looking for an electrical outlet for ato plug ofin an induction cooker.

Japanese outlets typically hasve two sockets basically.

My wife: How about plugging int into your nostrils?

The party was held without incidents.

"Incident" used in this context would be singular, as opposed to plural.

I had a barbecue party in the yard of my wife's parents' house in Shizuoka on the last weekend.

We had a hot pot besides barbecuesome barbecue and a hot pot.

I was looking for an electrical outlet for ato plug ofin an induction cooker.

Japanese outlet(For reference, Japanese sockets generally hasve two sockets basically.holes)

Here you should use a "parenthetical" - this sentence is not part of your story, it's a "pause" to give more information to the readers. This would be the most natural way to put this sentence into your story.

Also, here is a picture of an American outlet (i.imgur.com/dz0VtYx.png).

In that picture, there is 1 outlet, 2 sockets, and 6 holes. Hopefully this gives you an idea of the correct terms :)

My wife: How about plugging int into your nostrils?

Me: If the cooker turns on with them, I willelectricity came out of my nose I'd be happy to do so, anytime, anywhere.

Reworded to be more natural based on your native text.

The party was heldent off without incidentsa hitch.

This is a typical expression one would use in just this situation, to summarize what happened at an event. This expression is a more natural fit here.

I had a barbecue party in the yard of my wife's parents' house in Shizuoka on the last weekend.

We had a hot pot besides barbecueBesides barbecue, we also had a hot pot.

Or

Not only did we have barbecue, but also a hot pot.

I was looking for an electrical outlet for athe plug of an induction cooker.

I was looking for an outlet to plug the induction cooker in.
is also ok

Japanese outlets hasve two sockets basicallyholes(, by standard).

a socket is the entire thing :)

My wife: How about plugging int into your nostrils?

The party was held without inaccidents.