My mouse cursor Appear and disappear

There may be a built in Windows way to do it but this Lifehacker link suggests a tiny program called MouseJail to keep the mouse on the correct screen.

Try the mods, Better Dialogue Controls and Better MessageBox Controls.

Better Dialogue Controls description:

In A Nutshell

Have you ever pressed the E (activate) key in a dialogue and it selected something else than you expected? Then you may find this mod useful. Think of it as a patch for the controls of the dialogue interface.

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This mod will be most useful for people who like to use both the keyboard and mouse, it fixes issues where the mouse just flat out prevents some keyboard controls from working as expected. If you exclusively use the mouse to scroll and click through dialogues you may not have run into some of these issues. Likewise if you do not move the mouse at all after entering a dialogue. If you move the mouse pointer after entering a dialogue, then you will run into the issues below:

• A fairly common occurrence is when you leave the mouse pointer somewhere over the menu and you want to use the keyboard controls. If you like to switch to the keyboard navigation now and then, you will run into issues where the mouse pointer just flat out prevents you from scrolling to the top or bottom of the list with the keyboard controls.
• Another common occurrence is that as you enter the dialogue, you think the first item, next to the pointy bit, is the selected item. However when you press E, you find that the 2nd or 3rd item has been selected. This happens because the mouse pointer takes the focus away from the default option. Often times, I had to move the mouse pointer out of the way when entering the dialogues because of this.

Solution:

• The mouse focus now highlights items, but does not make them active. Thus the active item is always at the center of the list (next to the arrow/pointy bit). When you enter a dialogue, the mouse pointer never takes the focus away from the default option.
• The keyboard controls lets you scroll all the way to the first or last item, no matter if the mouse pointer is hanging over the menu.
• Scrolling the list with the mousewheel, or up/down controls always maintains the center item as the active one.
• You can still click off center dialogue items with the mouse, the mouse just won’t affect the menu until you click.
• The mousewheel scrolling has a smoother transition. It happens because I changed the mousewheel scrolling to work exactly as if you pressed the up/down controls. I didn’t plan to change the mouse wheel scrolling appearance, but I thought it was nice so left it in.

^{From: http://skyrim.nexusmods.com/mods/27371/}

Better MessageBox Controls description:

In A Nutshell

This mod lets you navigate the message boxes with the keyboard. Use Left/Right controls, and Activate to select options (typically A,D and E but it should work if you remapped those controls to other keys). Fixes the clickable area of buttons so they are easier to click.

Detailed Changes

These are the changes made to the vanilla (original) Skyrim message boxes:

• Enabled the gamepad style controls, which lets you navigate between options with the Left/Right keys (usually WASD), and select an option with the Activate key (usually E) and Return/Enter keys. Now you can dismiss simple “Ok” message boxes by pressing a single key instead of being forced to click a tiny text label.
• The TAB key cycles through "Exit" type buttons, where the label matches exactly "Return", "Exit", "Done", "Cancel", "Back" or "No". This helps navigate custom menus from mods that have lots of options. While SkyUI3/MCM will alleviate this, there are still many instances where mods will continue to use message boxes (or mods that won’t update to use MCM).
• With SKSE enabled pressing ESCAPE picks the first “exit” button and selects it. Thus you can quickly exit out of most dialogs that provide an “exit” button. The recognized “exit” buttons are exactly as for the TAB key (see above). You can quickly exit out of Yes/No dialogs, and mod options. In multi level mod options usually the ESCAPE key will take you to the parent set of options, because it picks “Return”, “Back” or “Cancel”, etc. WITHOUT SKSE the ESCAPE key behaves exactly as the TAB key (I’m sorry but it was not possible otherwise to distinguish the keys as crazy as it sounds!).
• Fixed the width of the clickable area for the buttons to properly adapt to the entire length of the button label.
• Extended the clickable area also below the button labels, instead of only above. This makes them easier to pick with the mouse.
• Added a subtle highlight to the focused button.

^{From: http://skyrim.nexusmods.com/mods/28170/}