Need correct results from 6 rss (fetch_feed)

What you’re looking for is probably something along the lines of:

<ul id="news-ticker" style="float: left;height: 15px !important; margin-left: 2px; margin-top: 10px; overflow: hidden; position: relative;width: 500px;">
<?php
    $rsslist = array(
            "http://thehoopdoctors.com/online2/newfeed/",
            "http://thepigskindoctors.com/newfeed/",
            "http://thepuckdoctors.com/newfeed/",
            "http://thecagedoctors.com/newfeed/",
            "http://thedugoutdoctors.com/newfeed/",
            "http://videogoneviral.com/newfeed/"
        );
    foreach ( $rsslist as $rssurl ): /* For every URL in the feed list */
        $feed = fetch_feed($rrsurl);
        foreach( $feed->get_items() as $item ): /* For every item */ ?>
            <li class="news"><a style="color:white !important;" href="https://wordpress.stackexchange.com/questions/38596/<?php echo $item->get_permalink();?>"><?php echo $item->get_title();?></a></li>
            <?php break; /* Do only once, we only need one item */ ?>
        <?php endforeach; ?>
    <?php endforeach; ?>
</ul>

Note how the code is fetching each URL separately instead of in multi-source mode and breaking out in the foreach $item loop as soon as an item is added. This should accomplish your task of getting one item from each feed in your ticker.

This code will do the same but grab each consecutive item from each feed:

<ul id="news-ticker" style="float: left;height: 15px !important; margin-left: 2px; margin-top: 10px; overflow: hidden; position: relative;width: 500px;">
<?php
    $rsslist = array(
            "http://thehoopdoctors.com/online2/newfeed/",
            "http://thepigskindoctors.com/newfeed/",
            "http://thepuckdoctors.com/newfeed/",
            "http://thecagedoctors.com/newfeed/",
            "http://thedugoutdoctors.com/newfeed/",
            "http://videogoneviral.com/newfeed/"
        );
    $feedlist = array();
    foreach ( $rsslist as $rssurl ) $feedlist []= fetch_feed( $rssurl ); /* store in feed array for later access */

    $current_item_cycle = 0;
    while ( sizeof($feedlist) ) { /* while feedlist is not empty */
        foreach ( $feedlist as $index => $feed ) { /* cycle through each feed */
            if (!$item = $feed->get_item($current_item_cycle++)) unset($feedlist[$index]); /* if feed has not more items unset */
            else /* echo it out */ { ?>
                <li class="news"><a style="color:white !important;" href="https://wordpress.stackexchange.com/questions/38596/<?php echo $item->get_permalink();?>"><?php echo $item->get_title();?></a></li>
            <?php }
        }
    } ?>
</ul>