Hydroponics (also known as soilless production) is a crop production system that does not use soil and the plants' fertiliser needs are supplied as a nutrient solution. This makes hydroponics a very efficient, resilient, flexible, sustainable and productive method of growing a wide range of crops.
All plants are organic in the true sense of the word (as opposed to rocks which are inorganic). The contemporary use of the term 'organic' refers to a method of product branding that specifically relates to a 'set of rules' for growing a crop and/or managing a cropping area.
There are a wide range of 'organic' brands for which there can be different rules but in general the key principle is that synthetic pesticides are not used (though some systems do permit the use of other pesticides). Often, the rules also require that a plant be grown in soil and nutrients are supplied in an organic form. Whether or not hydroponics can be 'organic' depends on the rules that have been decided on to govern the particular organic brand or accreditation scheme.
Hydroponics can successfully use nutrient sources that meet typical requirements of 'organic' systems and the decision to use (or not use) synthetic pesticides is a personal one and is not dependent on the type of production system being used, whether that be hydroponic, 'organic' or conventional.
No, not really. In terms of "no soil", plants can be found growing naturally in sand and gravel and even peat. Some plants known as epiphytes naturally don't grow "in" anything, but instead get water and nutrients from the moist air around them (in hydroponic, this is called air culture or aeroponics). Hydroponics can use a wide range of substrates. These can be naturally occurring inorganic materials such as sand and gravel, organic materials including coir fibre and peat, processed natural materials such perlite and rockwool (made from rock) as well as synthetic materials such as specialised foam.
As for the "nutrient solution", all plants (whether they are growing in the wild, in your organic garden or in a hydroponic system) take up nutrients through their roots as inorganic ions dissolved in water.
Organic fertilising materials such as compost and manure decompose to produce inorganic ions that become dissolved in the water around a plants' roots which the plant can then take up. Hydroponics simply fast-tracks this process by supplying the nutrients 'ready-to-use'.