Very well done. I liked your logic and you truly explained well along with the different elemental properties. But, as you explained only the clash of 2, I will go ahead and explain the rest, of what I believe to be necessary. This will allow you a very good knowledge over the logical interactions between the elements.
Fire - Water
If both fire and water will clash and negate one another, we can say a lot of steam will be created from this. The reason why it will be created is a simple reaction which is boiling this water, and changing it's state from liquid to gas, forming humidity in the air, which is a lot thicker than other gases not allowing you to see. It will be like a thick mist. The steam will mostly be thick enough for me to be able to maneuver around without the enemy seeing what I am doing. I could use this chance to either throw another attack, or change my location. Keeping in mind through, the enemy could do the very same thing, or just use a very strong water technique to blow the steam away. Another fact would be the sizzling sound, which might mask other sound movements, or terrain alterations in distance.
Fire - Wind
Fire and wind have a unique relation with one another, wind is able to make fire stronger if you combine it, but if they clash together, wind will take the disadvantage. The oxygen present in the wind will help the fire to burn stronger. Rule of thumb is, fire is one rank stronger than wind. So if you were to clash an A rank fire vs an A rank wind, fire will take over and beat the wind. Another thing is that if fire and wind are use both into conjuction, you could see that the fire will pick up in size, and block more vision of the enemy but just like that your own too. If they clash against one another, the fire will get weaker, getting smaller in size, but continuing. The chakra would burn more from the oxygen,but since you are wasting it faster, it will be a lot weaker than before. Smell wouldn't be much different, you can't see sth too great, you could hear, the fire being amplified and the wind flow. You know how you blow in a flame? This sound will be of aid. Another one is the limiting view you could get from that certain situation. And the look how the fire will be a lot brighter. If wind and fire was to clash, wouldn't you think the direction of the heat be manipulated slightly, even if it is overpowered? As well as the light and flares of the fire? This can help you find out the direction of the wind blast and allow for mobility. If a fireball is clashing with a wind sphere, the sphere is sent straight and hits it directly, the fireballs heat and flares would momentarily be pushed directly back, nothing to harm you, it just happens logically. So you can easily find out the sphere had a straight trajectory and can deduce the opponent will probably be straight ahead of you. This can be a neat way of sensing everything, I wouldn't pull it out a lot, but against fast opponents or stealthy people, it can help you have a excuse of possible mobility. Also, imagine dust clashing with fire? You can probably hear the cracking and bounce of dust and debris being burned from the flames, small sparks and ash linger around the air, can take advantage for hiding or adding to attack. Other then that, good job.
Fire-Earth
Fire and Earth have not a noticeable relation against one another, rule of thumb is that they are both equal, A fire = A Earth. But where these 2 elements shine is combos. The unique forms of earth, mud being one of them can be used perfectly well with fire. Using fire in conjunction with this mud, you can create some kind of pseudo-Lava. It will be molted earth, hot enough to burn the flesh to the bone, and fairly well block the view or movements. Another one is if you use solid earth, along with fire, you can create a similar fashion to a meteor. The moment this rock will shatter, the fire will spread burning. It would though, be deceived because of the unusual glow which earth can make if you amplify it with fire. The smell would certainly be different from normal. When normal earth or mineral burns, the different kinds of odors released could trigger something. The sight would be as I said, very distinctive no matter how you look at it, it will glow in a red color, and by the looks it will look more liquidy. You know?
Fire - Lightning
Lightning and fire are arguably the 2 strongest elements because of their raw destructive power. But they never happen to crash together considering both of them are energy based, they will phase through one another. But let us take it in other ways. We knows fire covers a huge more vision area than lightning, while lightning might move in either projectiles or bolts. So, when you clash together a very large fire stream against a bolted lightning technique, the person which used the bolt of lightning will lose the enemy from their sight, the attack might not match entirely. The sounds will continue respectively, maybe eaten up a little from the rough continuing noises of fire. And the scent won't change much, as they don't interact.
Water - Wind
Once water and wind clash together negating, we could say there will be large amounts of water still present in the battlefield. But due to the force, they hit one another, they would naturally have the water explode in tiny droplets, falling mostly everywhere in the battlefield, maybe a lot less towards the location where the wind hit it. The water will also have a large amount of tribulation, so high amounts of water might be colored white, due to the pressure and intensity of the attack, maybe even steam in rare cases. Though, the water would be good enough. Naturally in those occasions, the enemy would squint, or cover his eyes for the water not to momentarily blind him. Another thing which might happen is the battlefield getting wet, if that were in glass we would observe slippery, and a high lack of dust or small sand particles, as they now are more mud like.
Water - Lightning
Once lightning and water collide in most occasions the lightning would simply move through the water, using it as a conductor, rather than just destroying it. Might be perfect if the enemy is using a stream based attack, and you could kill him, but when it is a projectile one, best used would be raw, powerful and destructive lighting, generally found in compressed forms like projectiles, or attacks connected to the body, tangible ones like the chidori spear. Once those collide, generally the lightning and the high amounts of heat, would evaporate amounts of water, not all of it of course. The rest would simply fall in small particles, like rain or larger chunks as well. The steam along with the other rain drops prove themselves amazing by blocking sight, or other senses. There will surely be sizzling sounds, as well as lightning discharge sounds because of the shortcut created into water. Another positive thing is that the small sand grains become unable to be used, and the mist could cover movements.
Water - Earth
Water and Earth generally comes with the win of earth in most occasions. Using the earth's ability to not let water move through along with it's sturdiness which rivals water, there is no real discussion who might win, but if they crash and negate many things would be visible. Unlike before, where water clashed non tangible elements generally, earth will now, along with water break. The earth chunks would not be in the size of grains, but medium sized rocks, which might prove themselves harmful if they hit too hard. Another plausible thing would be, that since the water is hitting a chance enforced earth, they might even create slight appearances of mud, along with the rocks and normal water rain. The sound of the rain drops, the heavier drops of rocks and debris, along with the turbulance that the water has, would not only hinder the sight, but also the hearing.
Wind - Earth
When wind and earth clash together, most of the time wind comes superior. In cases when they negate one another, the strong forces of wind, will most certainly shatter most very strong and sturdy earth constructs, which would send flying a lot of debris, and depending on the strength of both techniques, these chucks can be enough to send someone K.O, or they could be small enough to render vision badly. Also, depending on what direction the wind was going, the earth will mostly go towards that, with a slight omnidirectional disperse. If it was the forms of mud, the mud will most likely do the same, which is fly in sticky chunks in many locations all around the present location where it was on.
Wind - Lightning
There is not a lot I could say here specifically, because in many cases, lightning and wind simply causes the disperse of lightning, but think about it. Solid or at least, shaped lightning constructs will force the wind to break it down first, so there will be thousands of lightning sparks, as the lightning goes from stable, to unstable, and logically thinking, as the lightning goes to unstable there will be a lot of sound, and very powerful sparks of light.
Any questions to far?