Role of solvents in ozonolysis and oz0nolysis of alkynes with water
Ozonolysis:

Ozone, an allotrope of oxygen, is highly reactive molecule that can undergo a 1,3-dipolar addition across alkenes, forming a 1,2,3-trioxolanes. This compound is unstable and undergoes a cycloreversion to form a carbonyl compound and a carbonyl oxide. These resultant compounds (the carbonyl compound and the carbonyl oxide) then proceed through a second 1,3-dipolar addition to form a 1,2,4-trioxolane, commonly known as an ozonide. To complete the cleavage of the alkene, the ozonolysis (implies that ozone causes to break (-lysis)) is followed by a work-up designed to break apart the ozonide.  The work-up, which can be oxidative or reductive, determines the oxidation state of the products. For example, cis- or trans-alkenes gives two corresponding aldehydes with reductive work-up while same alkene gives two corresponding carboxylic with oxidative work-up.

The most common reductive work up procedure uses dimethyl sulphide (DMS) in the
presence of methanol. Methanol reacts with the carbonyl oxide to form a hydroperoxy acetal, and this is in turn reduced by the dimethyl sulphide to give relevant carbonyl compound and dimethyl sulfoxide. It is also possible for dimethyl sulphide to reduce the ozonide without the presence of methanol as well, although reation would proceed slowly (A new and convenient method for converting olefins to aldehydes: Tetrahedron Letters, 1966, 7(36), 4273–4278). Other reductive methods: $\ce {Zn/AcOH}$, triphenylphosphine ($\ce {PPh3}$), or thiourea ($\ce {NH2C(=S)NH2}$).

Note: Recently, there is a  publications claiming an efficient and convenient method for reductive workup of the ozonolysis reaction using sodium hydrosulfite. However, the final product could be either aldehyde or acid, thus the method is not selective.

Oxidation: The direct conversion of terminal and 1,2-disubstituted alkenes to carboxylic acids is described as oxidative ozonolysis. This can be easily achieved through treatment of initial ozonolysis products with hydrogen peroxide ($\ce {H2O2}$) or other oxidants.

You may also find answers to your Q3 and Q4 from the reference given above.