How can I resize the count badge?

You may adjust font-size, like this:

div.da-reactions-container div.reactions div.reaction .count {
    font-size: 0.3em;

Using the complete hierarchy in selector give the right priority, but event this will have the same effect:

.count {
    font-size: 0.3em !important;

It is important to avoid the !important keyword because it could bring to unexpected results, moreover a complete hierarchical selector is good to avoid collision with other elements in page.

With this selector you may change everything, for example you may display numbers under images:

div.da-reactions-container div.reactions div.reaction .count {
    position: relative;
    right: 0;
    line-height: 1em;
    background: transparent;

You can probably add additional CSS directly through WordPress interface if your Theme supports this feature, it should be at Appearance » Customize » Additional CSS.

If your theme does not support this feature you can always write those CSS lines in your main CSS file.