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I looked at the documentation. Within the documentation scroll down to the "Creating Taxonomies" section and set-up the taxonomies if you haven't done this already. The taxonomies are Status, Types, Price, Locations and Bedrooms. See if this allows you to import the missing items.
Have you installed the AgentPress Listings plugin? This plugin likely creates any necessary post types that go with this theme. If the post types aren't in place before try to import the sample content, then parts of the import will fail.
If you are referring to comment spam, you can check out Akismet. If you are referring to contact form spam, some form builder plugins such as Gravity Forms have spam protection options built in. If you don't own a Gravity Forms license, Ninja Forms is worthing checking out - they have a free version. I haven't had a chance to try out Ninja Forms, but I hear good things about it.
For security, I typically lock my sites down with iThemes Security Pro, which is a premium plugin. But they also have a free version available.
To improve performance, I typically use W3 Total Cache.
Hope this helps.
I haven't used that theme before. But I think it requires the AgentPress Listings plugin.
The slider image size for this theme is 1140 by 445. I typically size the image to the correct dimensions before uploading them. It produces the best results. Otherwise WordPress will try to crop the image for you. Your images may be getting cut off, because they are too small for WordPress to crop to the correct size.