What is the state of aggregation (gas, liquid) of oxygen in blood?
Regarding the state of oxygen in blood: It is in solution in the blood plasma (which mostly consists of water), in the form of single molecules. Think of water which you leave exposed to air: carbon dioxide will be captured and dissolved (along with the other gases in air), but these molecules are not gaseous or liquid, but rather "in solution", which is different from the "classical" states.

Back to oxygen: As your reference already states, most of the oxygen in solution will bind to hemoglobin. The actual state of oxygen in that complex has been debated, but it is believed to be reduced by the hemoglobin iron to the superoxide anion, coordinated to Fe$^{3+}$. See Wikipedia on this.

Also, the lungs do not "convert" the atmospheric oxygen to anything, they rather allow, due to their very large surface area, the quick exchange of oxygen/carbon dioxide in solution and in the air.