First, on the difference between あり and あって: The two perform the same function here (transitioning to the next clause with "and"), but using the ～ます stem form as a transition is more literary than the ～て form. In speech, you're far more likely to hear あって.
Now as for where these come from and where else they can be used, let's look at nouns as an example. Recall that である can be used in place of だ:
彼は監督である。 He is a manager.
Now in modern Japanese, you wouldn't hear this in speech, because である has been relegated to the status of "literary construction". But what if you wanted to say, "He is both a manager and a player."? You might be tempted to say something like this:
Unfortunately, this is incorrect. To fix this, we use である behind each noun, with the も particle in between the で and the ある:
彼は監督でもあり、選手でもある。 He is both a manager and a player.
(In speech, you would replace the あり with あって, but the last ある will stay the same.)
And since we can treat the ある in である just like the regular verb ある (some grammatical purists may argue that it's not in the same category, but oh well), we can apply this strategy to negatives:
彼は監督でもなく、選手でもない。 He is neither a manager nor a player.
(And as expected, in speech you would replace なく with なくて.)
By extension, the same rule applied to adjectives (you've already seen the positive form):
この本は面白くもなく、つまらなくもない。 This book is neither interesting nor boring. (How odd!)
Incidentally, the particle は can substitute for も with the above adjective example. In this case, it indicates a point on a scale somewhere in between the adjective used and its opposite.
この本は面白くはない。 This book [may not be totally boring, but it] is not interesting.
Simply saying 面白くない typically implies that it's つまらない, but saying 面白くはない implies that it doesn't quite reach all the way to the point of being つまらない.