Splitting the Earth's emerged land in two equal parts
TL;DR: It depends on how one defines land area. With a sensible definition, such a meridian (along with its completion on the other side of the globe) must necessarily exist by virtue of the intermediate value theorem. The same goes for the line of latitude.



Defining the functions


$\operatorname{A}_{\text{E}}(\lambda)$ is the area of the land projected onto the reference ellipsoid of the hemisphere to the east of the line of longitude $\lambda$.
$\operatorname{A}_{\text{W}}(\lambda)$ is the area of the land projected onto the reference ellipsoid of the hemisphere to the west of the line of longitude $\lambda$.
$\operatorname{\Delta A}_{\text{lon}}(\lambda)$ is the difference between $\operatorname{A}_{\text{E}}(\lambda)$ and $\operatorname{A}_{\text{W}}(\lambda)$.
$\operatorname{A}_{\text{N}}(\phi)$ is the area of the land projected onto the reference ellipsoid of the hemisphere to the north of the line of latitude $\phi$.
$\operatorname{A}_{\text{S}}(\phi)$ is the area of the land projected onto the reference ellipsoid of the hemisphere to the west of the line of longitude $\phi$.
$\operatorname{\Delta A}_{\text{lat}}(\phi)$ is the difference between $\operatorname{A}_{\text{N}}(\phi)$ and $\operatorname{A}_{\text{S}}(\phi)$.


My definitions make each of these functions continuous. Note well: Other definitions of land area might result in non-continuous functions. Imagine a perfectly vertical cliff that runs north to south. If the area of that cliff face counted as land area then $\operatorname{\Delta A}_{\text{lon}}(\lambda)$ will not be continuous. The same goes for $\operatorname{\Delta A}_{\text{lat}}(\phi)$ for a perfectly vertical cliff that run east to west.

Proving that the latitude line must exist is easy, so I'll do that first. All of the Earth's land area is north of 90° south latitude, making $\operatorname{\Delta A}_{\text{lat}}(-90)$ a large positive number. All of the Earth's land area is south of 90° north latitude, making $\operatorname{\Delta A}_{\text{lat}}(90)$ a large negative number. Because zero is between this large negative number and large positive number, and because $\operatorname{\Delta A}_{\text{lat}}(\phi)$ is continuous, there must necessarily exist at least one line of latitude $\phi$ for which $\operatorname{\Delta A}_{\text{lat}}(\phi)$ is zero.

Regarding longitude, Pick an arbitrary longitude $\lambda$. If $\operatorname{\Delta A}_{\text{lon}}(\lambda)$ is zero we have a winner. If it's not zero, then since $\operatorname{\Delta A}_{\text{lon}}(\lambda +180°) = -\operatorname{\Delta A}_{\text{lon}}(\lambda)$, there exists at least one longitude $\lambda_0$ between $\lambda$ and $\lambda+180°$ where $\operatorname{\Delta A}_{\text{lon}}(\lambda_0)$ is zero.