widget coding issue

I’m trying to learn how to create widgets and I have 2 questions:
first… where can I put my sidebar widget file so that wordpress will load them. I’d like to keep the widget code out of the function file. I tried putting it in it’s own file under wp-content/plugins/widget-name/widget-name.php but that did not work. I would like to make it available to “all” my themes so I did not want to put it in a widget folder under a specific theme theme-name/widgets/widget-name.php as I had read was possible. Is there somewhere I can place my widget-name.php file top make it available to all of my themes?

Second question is a frustrating coding issue. Could anyone tells me why the following code:

<blockquote>function form($instance){
		?>
		<p>
			<label for="<?php echo $this->get_field_id('title'); ?>">
			<?php _e('Title:'); ?></label>
			<input id="<?php echo $this->get_field_id('title'); ?>"
				name="<?php echo $this->get_field_name('title'); ?>"
				type="text"
				value="<?php echo $instance['title']; ?>" />
		</p>
	 	<?php
	}</blockquote>

gives me a “Notice: Undefined index: title” in my widget input box when I first place it in the side bar. but the following code does not:

<blockquote>function form($instance) {
		echo '<p>';
		echo '<label for="' . $this->get_field_id("title") .'">Random Trivia:</label>';
		echo '<input type="text"';
		echo 'name="' . $this->get_field_name("title") . '" ';
		echo 'id="' . $this->get_field_id("title") . '" ';
		echo 'value="' . $instance["title"] . '" />';
		echo '</p>';
	}</blockquote>

(they seem nearly identical to me)?

Thanks for any help