Arabic word breaks when changing a letter’s weight using a variable font. How can I keep the letters connected when using variable features?
Of course. Different style runs will be treated like different fonts. That is how type works in Adobe InDesign. You have to change the whole word.
Is there away we can overcome this? I have a need (without going into too much detail of this university project) to mix styles in each Arabic word of the text. So, I need to overcome how OpenType forbids joined letters of mixed styles.
OpenType keeps breaking my words 
Is there an OpenType feature that can allow me to supersede OpenType’s breaking of the word into isolated letters? or is there away to make changes in the font table after I export?
No, this is not possible. Every style is a different font and different fonts cannot interact with features. For example, kerning is also not possible across styles.
A solution would be to have thin and bold glyphs in the same font and then change the selected glyphs using a style set feature. This is possible, but it requires a lot of work and also makes switching styles more difficult for the user. And then you also need to design custom glyphs that connect a thin to a bold letter and vice versa and write custom feature code to handle these transitions.
There’s a chance if you use it in a software where you can force the positional feature. No guarantee that works though. And between a bold and a light style, the connections are unlikely to match anyway. It will not look good.
You might be able to achieve what you seek by copying the Unicode characters of the individual letters according to their positions.
ﺃ ﻬ ﻼ - ﻫ
ﺃ ﻫ ﻼ
You can use MS word to insert these characters and then copy them.