VS 2017 Express
Form で作成、.Net Framework 4.6を指定した。
app.ico が別途必要。https://icon-icons.com/ja/ などからアイコンを準備

Program.cs

using System;
using System.Diagnostics; // ProcessStartInfoで必要
using System.Drawing;
using System.Windows.Forms;

class MainWindow
{
	static void Main()
	{
		Launcher launcher = new Launcher();
		Application.Run();
	}
}

class Launcher : Form
{
	public Launcher()
	{
		this.ShowInTaskbar = false;
		this.setComponents();
	}

	private void cmdHello()
	{
		MessageBox.Show("こんにちは。");
	}

	private void cmdIe()
	{
		ProcessStartInfo processStartInfo = new ProcessStartInfo("C:\\Program Files (x86)\\Internet Explorer\\iexplore.exe", "http://yahoo.co.jp");
		Process process = Process.Start(processStartInfo);
	}

	private void Close_Click()
	{
		Application.Exit();
	}

	private void setComponents()
	{
		NotifyIcon icon = new NotifyIcon();
		icon.Icon = new Icon("app.ico");
		icon.Visible = true;
		icon.Text = "常駐アプリテスト";
		ContextMenuStrip menu = new ContextMenuStrip();
		ToolStripMenuItem menuItem = new ToolStripMenuItem();

		menu.Items.AddRange(new ToolStripMenuItem[]{
			new ToolStripMenuItem("命令cmdHello", null, (s,e)=>{cmdHello();}),
			new ToolStripMenuItem("命令cmdIe", null, (s,e)=>{cmdIe();}),
			new ToolStripMenuItem("終了", null, (s,e)=>{Close_Click();})
		});

		icon.ContextMenuStrip = menu;
	}
}

いろんなページを参考にしました。