Grignard reagents are very strong bases, and therefore can be spoiled by protons. Grignard reagents are so basic in fact that any protic solvent will render them useless. Dimethyl ether is the only non-protic solvent, and is therefore the correct answer. 

The molecule pictured above is known as an ether because it contains an oxygen atom within the sequence of a carbon chain. (An oxygen atom bonded to two carbons in a carbon chain).

Therefore the correct answer is ether. All other answer choices are carbonyls, meaning that they contain a carbon atom double bonded to an oxygen atom. 

IV>III>I>II

The most acidic compound is option IV because it contains a carboxylic acid group which is the most acidic organic functional group. Phenols are weak acids. Alcohols in solution are slightly less acidic than water and therefore are slightly basic. 

The molecule's longest carbon chain has 5 carbons (thus, "pent"), and the one double bond makes it an alkENE (thus "pentene"). The longest chain is a ring structure (thus "cyclopentene"). Because the IUPAC rules automatically assign the location of the first double bond to carbons 1 and 2, there is no need for a number locand.

The molecule's longest carbon chain has 4 carbons (thus, "but"), and the presence of two double bonds makes it an alkENE, more specifically, a diene (thus "butadiene"). Because there is more than one way in which the double bonds can be arranged (between C1-C2 and C2-C3, or between C1-C2 and C3-C4), it's important to place locants indicating the lower-numbered carbon in each double bond.

The molecule's longest carbon chain has 6 carbons (thus, "hex-"), and the lack of double bonds makes it an alkane (thus "hexane"). The longest chain is a ring structure (thus "cyclohexane"), and the one branching group is a carbon chain consisting of one carbon and no double bonds (a "methyl" group). Because there are no other functional groups on the molecule there is no need to put a number before the location of the methyl group (thus "methylcyclohexane").

The molecule's longest carbon chain has 5 carbons (thus, "pent"), and the lack of double bonds makes it an alkane (thus "pentane"). The one functional group is a bromine atom attached to carbon number 3 (whether read from left to right or right to left, the bromine is always on carbon number 3). Thus, the molecule is named "3-bromopentane."

The molecule's longest carbon chain has 5 carbons (thus, "pent-"), and the carbon-carbon double bond makes it an alkENE (thus "pentene"). The location of the double bond must be specified, and numbering the carbon chain to give the double bond the lowest numbers possible mean that it is numbered from right to left, putting the double bond between carbon 2 and carbon 3. This will put the methyl group on carbon 3. 

Regarding stereochemistry, on carbon 2, the higher priority substituent is the methyl group. On carbon 3, the ethyl group is the higher priority. The higher priority substituents are on the same side of the double bond, and therefore the stereochemistry designation is "Z."

The molecule's longest carbon chain has 6 carbons (thus, "hex-"), and the lack of carbon-carbon double bonds makes it an alkANE (thus "hexan-"). The presence of a hydroxyl group makes this molecule an alcohol (thus "hexanol"). The longest carbon chain is a ring structure (thus "cyclohexanol"), and the location of the alcohol group is assumed to be carbon 1 because it's the highest priority functional group on the molecule. The only other substituent is a methyl group, and numbering the carbon chain starting from the one containing the alcohol group and moving toward the methyl group puts the methyl group on carbon 2. Thus "2-methylcyclohexanol."

The molecule's longest carbon chain has 6 carbons (thus, "hex-"), and the presence of three double bonds makes it an alkENE, more specifically, a triene (thus "hexatriene"). Because there is more than one way in which the double bonds can be arranged it's important to place locants indicating the lower-numbered carbon in each double bond (1, 3, and 5 in this case).