Custom Logging Class: Efficiency
Efficiency


the biggest improvement will be to not open the config file each time you log something, but only once when instantiating the logger.
you could also open the log file only once, but it might not be a good idea to have it open for the whole lifetime of the program. 
create only one instance of the two SimpleDateFormat objects and store it in a class level field instead of creating them each time.
also create your messages array only once as a field.


Robustness


use an enum instead of string for your warning levels
you are never closing your propFile stream.


Readability


follow the standard Java indent style: curly brackets go on the same line as the opening statement.
use more spaces (around &lt;=, =, etc). This point and the previous one can be fixed by any IDE.
don't do this: timeStamp+id+++"|".
use camelCase for all variable names, don't mix it with snake_case except for final static fields.
use JavaDoc style comments.
remove variables that are only used in one place.
use try with resources to save the close statement.
choose better variable names. app would be clearer as shortDateFormat, format as extendedDateFormat, dateApp as logFileName, appendValue as appendToFile, logLevel as maximalLogLevel (should this maybe be minimal?), logType as logLevel.


There are probably more improvements, but with this your code already looks a bit nicer:

public class FunctionLogging {

    private String path;
    private boolean appendToFile = false;
    private int logLevel;
    private String[] messages = new String[] {"FATAL", "WARNING", "NOTICE", "DEBUG"};

    private SimpleDateFormat format = new SimpleDateFormat("dd-MM-yyyy|HH:mm:ss:SSSSSS|");
    private SimpleDateFormat app = new SimpleDateFormat("dd-MM-yyyy");

    /**
     *
     * @param filePath used as a placeholder variable for the output path. Can be hard-coded but is preferredly configurable from properties file
     * @param logLevel set as true when called in each class used in. true appends line to the end of the file, false overwrites file contents
     * @param append_value TODO
     */
    public FunctionLogging(String filePath, int logLevel, boolean appendValue) throws IOException {
        path = filePath;
        appendToFile = appendValue;

        loadConfig();
    }

    private void loadConfig() throws IOException {
        try (FileInputStream propFile = new FileInputStream("config.ini")) {
            Properties config = new Properties(System.getProperties());
            config.load(propFile);
            logLevel = Integer.parseInt(config.getProperty("loggingLevel"));
        }
    }

    public void writeToFile(int logType, String textLine) throws IOException {
        if (logType &lt;= logLevel) {
            Date date = new Date();
            String timeStamp = format.format(date);
            String dateApp = app.format(date);

            FileWriter logWriter = new FileWriter(path + dateApp + ".log", appendToFile);
            try (PrintWriter loggerLines = new PrintWriter(logWriter)) {
                loggerLines.println(timeStamp + "1|" + messages[logType] + "|" + textLine);
            }
        }
    }
}


Or with enums:

public class FunctionLogging {

    public enum Level {

        FATAL("FATAL", 1),
        WARNING("WARNING", 2),
        NOTICE("NOTICE", 3),
        DEBUG("DEBUG", 4);

        private final String message;
        private final int level;

        private Level(String message, int level) {
            this.message = message;
            this.level = level;
        }

        @Override
        public String toString() {
            return message;
        }

        public int getLevel() {
            return level;
        }
    }

    // [...]

    public void writeToFile(Level level, String textLine) throws IOException {
        if (level.getLevel() &lt;= logLevel) {
            // [...]
            try (PrintWriter loggerLines = new PrintWriter(logWriter)) {
                loggerLines.println(timeStamp + "1|" + level.toString() + "|" + textLine);
            }
        }
    }

    // [...]

// usage:

writeToFile(FunctionLogging.Level.DEBUG, "something happened");