Before you concern yourself with this, you need to worry about other things:
a) Your page takes too long to load, because it's displaying too many images, and some of them are huge files. For example, the dog image I checked is 3888px wide. There's no point uploading a 3888px image when you display it at 630px: loading time apart, you're wasting your storage space, and you get worse quality on the post.
The one thing to do is use an image editing application or an online tool to downsize your images to the pixel width you want on the post, and upload and insert those versions. The other thing to do is truncate the posts using the read-more tag, and/or limit the number of posts per page to only a few. Before I tell you more about this last option: do you like the infinite scrolling feature that kicks in when you scroll to the bottom, or would you like to get rid of it?
b) The images in your widgets are also too large, and they've been uploaded to photobucket instead of your blog. This again means more loading time, risk of a no-show if something goes wrong in photobucket, plus some versions of Explorer may not understand the scaled-down version the theme imposes (that is, they may display the images in their original size or distorted). So again you should downsize these images to the right pixel width for your sidebar, then upload them to your blog via Media > Add New, copy the File URLs and use those URLs in the widgets.
As for the border/bg issue, yes of course they can be removed: CSS editing allows you to modify the appearance of any element.
But in this case you don't even need to tamper with the CSS. That frame is the default look of captioned images; you can get rid of it by simply using Text widgets instead of Image widgets. Coding here: