of course you can place a link whereever you want. only if we knew what you refer to as "blog menu"

regarding the search field, adding the below to the very end of your css file would suffice, or find those lines in your css file.

input.SearchField {
width:120px;/*change the width as you wish*/
}

a link to your website would help better

I am talking about the Blog home link (Menu link) widget. Can I add a link to the forum to it?
Thank you.

Just add a widget to the same containter as Blog home link, and add the html code for the link to your forum.

Under Menu add Free HTML Widget
Next enter something like this
<li class="widget_core_user_tools page_item"><a href="http://yoururl/youblog.php/forum/" title="Go to the Forum...">Forum</a></li>

Obviously replace proper urls in above
Don't put anything for Title...just fill in the content part
basically your telling it to use the style class from the widget_core_user tools_page_item i.e. your menu button and apply it to your link...
Now your link will look like the menu button...hope this helps