For the Kanji 校 is the fifth stroke connected to the sixth stroke? Some websites online show the fifth stroke as a vertical line straight down connected to the sixth stroke, while others (and in Chinese) have it as a downward diagonal dash. Is there are difference between the two? Similarly, is the first stroke in connected to the second or a diagonal dash? Is this always the case whenever a Kanji has something similar to these two?

It's the difference between printing, handwriting, and calligraphy.

This element, called nabebuta or keisankanmuri (), should be connected and vertical for more formal (printed) styles, and will normally be disconnected and 'diagonal' for calligraphic styles and many handwritten styles.

There is not right or wrong unless you are talking about a particular font style.

When practicing kanji early on, it is normally recommended to stick to a more 'standard' style, like Kyokashotai (thanks @drooze).