At the risk of losing rep, I'm going to go against policy and instead try to dissuade you from embarking on this challenge.
All usability experts agree that websites should make no sounds. It makes for poor website UX. Period.
Notifications should be handled visually only. Or, send the user a notification email and hope their email makes a noise when new messages arrive!
If you really want a reason to make notification sounds, program an app, and put them in there.
Negative Nancy aside, I'll take a stab at answering your questions.
What things should I take into account when creating these?
As with all design, aesthetics would be good to take into account. The world doesn't need any more beautiful/ugly sounds, depending on your POV. Remember, there is no accounting for taste, so at the very least, keep it simple!
Take into account your target demographic - for instance, as people age, their high end frequency hearing drops off. If you only want younglings to hear your sounds, keep it above 15K... :p
I would also take into account the huge catalog of notification sounds that already exist. Do you really need to roll your own? Are you certain you'll actually create something unique? Let us not reinvent the wheel. I found a version of ICQ's Uh Oh notification and use that in my messaging app. IMO nothing is better but I believe that opinion has attributed to it good memories (another thing to take into account).
Which software would I be best off using?
I would use software that has sound generation built in. Unless you plan on recording everything from scratch via a microphone, it would behoove you to get something with as wide a palette as possible. If you're on the Mac, I recommend Logic Pro X, as it has built into it a vast number of sound generating devices & effects.
Why use certain sounds?
Again, it depends on aesthetics. Sound can be mystical. Jarring can turn people off from your notifications. Soothing ones can help them look forward to hearing from you.
Or are you trying to wake people up?
Take some time and investigate what companies like Apple use for their notification sounds. They spend a great deal of R&D on these aesthetic decisions which you can use to your advantage by seeing what makes them tick. What do they have in common? How long are they? What kind of effects do they use? Are they electronic or organic in nature?
Since you are outside the sound/music industry, I recommend apprenticing with someone who does sound design for a living. Or intern at a local recording studio. Immerse yourself in that which interests you and it will invariably rub off.