Alt text is for visually impaired users. A screen reading device literally reads that text to them. It just happens that search bots read it too. It doesn't matter if it doesn't appear on screen.
If you do want some text to appear, add a title to your img tag.
<img alt="Workhorse Painting Contractor of Jacksonville Fl." title="My Text Here" srcset="./img/Workhorse3.gif">
Keep in mind, there are no mouseovers on phones so don't rely on them for important info.
Also, alt text should describe the image, this isn't how it's meant to be used:
<img alt="Get New Interior, Exterior, Paint with Workhorse Painting Co. of Jacksonville, Fl" srcset="./img/trademark.gif">
This doesn't tell a screen reader what the image is, and Google might see that as more spammy than helpful.
<img alt="galloping horse trademark" srcset="./img/workhorse-trademark.gif"> <<< Better
Your image doesn't have to be a link to display a title. If you have those images at the top of each page, you should link them to your home page since it's common to use a logo as the home page link. Empty links are just confusing – they make users think something is broken.
As for images that are just background graphics, you can remove the alt="" from the img tag so readers will skip those images. If a graphic is relevant to the content, go ahead and describe it. For example, some sites have icons at the top of content sections rather than text titles. To screen readers, these are images so say what they represent:
alt="Business Hours" srcset="./img/clock-icon.png"