OK, first of all and so we can help you most appropriately:
You say you are new to Moodle, but you are a designer - can I ask, graphic designer? web designer? what is your experience of css/php, using browser development tools etc? Have you looked at any of the Theme FAQs or HowTo pages linked from the top of the forum? - they are always a good place to start
You also say you don't know what version of Moodle you are running, yet if you take a look at the first line of the screenshot you have provided it clearly states you are running Moodle2.4.6+
OK, so some simple steps that can be expanded on if you need more detail:
1. Either, use a developer tool to identify the css selector for the white heading (I think its h1.headermain but I dont have 2.4 and Boxxie on my laptop with me to confirm) and apply display:none OR in the layout file remove/comment out the line that says <h1 class="headermain"><?php echo $PAGE->heading ?></h1>
2. Create page resources (Add an Activity/Resource>Page) and then use the custommenu to link to those pages. These can be created off the front page if you want, but I often create a separate course (with full guest access) to manage them separately
3. Edit the general.php file to add an image with a link where you want it - this is one of the areas where it would be helpful to know your experience/comfort level to provide instructions but in the first instance I would refer you to the how to pages https://docs.moodle.org/dev/Using_images_in_a_theme
4. Take a look at an existing non-bootstrap theme that incororpates a slider, there are a few in the themes database such as krystle (it needs to be a non-bootrap one in order for you to reuse the code as it is on Moodle2.4
5. No, upgrading your site will not lose all your data - but make sure you have things like your theme backed up as you may need to reinstall your theme as it is not included in 2.7 or 2.8. Also please note the repeated advice throughout these forums that code changes to what were core themes can be lost in an upgrade and it is beneficial to create either a clone or a child theme to make your changes to rather than changing boxxie directly!
For moving your site to a live server see https://docs.moodle.org/27/en/Moodle_migration
Hope that can help you start in the right direction and that you are able to come back with any specific questions to help with
Richard
