why is it skipping lines?

[PredicateNormative]

Because you need to clear the cin buffer of invalid characters:

#include <string>
#include <iostream>
#include <iomanip>
#include <cmath>
#include <fstream>
#include <cctype> // for tolower
#include <limits>   //for numeric_limits

using namespace std;

void selection();
void addChecksum();
void runChecksum();

int main()
{
    selection();
    
    // A more portable version of system("pause") is:
    std::cout << "Press return to continue" << std::endl;
    std::cin.get();
    return 0;
}

void selection()
{
    char choice;

    cout << "Please Select: " << endl;
    cout << "  A) Add checksum to specified file" << endl;
    cout << "  B) Verify integrity of specified file" << endl;
    cout << "  Q) Quit" << endl;

    cin >> choice;
    cout << endl;

    //Clear anything currently in the cin buffer
    std::cin.clear();
    std::cin.ignore(std::numeric_limits<std::streamsize>::max(), '\n');

    //You can reduce the size of your switch statement by using tolower.
    [COLOR=Red]switch (tolower(choice))
    {
    case 'a':
        addChecksum();
        break;
    case 'b':
        runChecksum();
        break;
    case 'q':
        return;
    default:
        cout << "Your choice did not match any of the menu options." << endl;
    }
}

void addChecksum()
{
    string inputFileName;
    ifstream inputFile;
    string line;

    cout << "Specify the file path: ";
    getline(cin, inputFileName);

    inputFile.open(inputFileName.c_str());
    cout << endl;

    while(!inputFile.is_open())
    {
        cout << "The file " << inputFileName << "could not be found or opened!" << endl;
        //You have a bug here. What if someone realises they don't
        //want to load a file anymore, but want to quit instead?
        //selection will keep being called even though the user is
        // pressing 'q' because they have not entered a valid file name!
        selection();
    }
}

void runChecksum()
{
    string inputFileName;
    ifstream inputFile;
    string line;
    int checksum;

    checksum = 0;

    while(!inputFile.is_open())
    {
        cout << "Specify the file path: " << inputFileName;
        getline(cin, inputFileName);

        inputFile.open(inputFileName.c_str());
    }

    //I guess you are going to do your calulation here?
    // ....
    cout << "File checksum = " << checksum << endl;
}