I like playing mages, but I have to agree with the general consensus of this site that the vanilla (unmodded) magic system leaves a little to be desired. Nevertheless, my first and only playthrough on Master difficulty was with a pure mage so it is possible.
1.) Bound Weapons - will the damage they do increase as your level of Conjuration increases? Do you need to spend perk points on things like One-handed and Archery to make these more effective? (Seems counter-intuitive to do that if you're trying for a pure mage).
Your Conjuration skill doesn't directly affect the damage done by bound weapons, but the Mystic Binding perk does. You can get it relatively early, but a common complaint is that bound weapons are underpowered later in the game. Mods can fix this.
And yes, weapon skills and perks do affect Bound Weapon damage, so they are not the best choice for a "pure" mage, as you will have to level those skills somewhat.
2.) Staffs - will using staffs help train up whichever magic category they represent? Will a Staff of Fireballs train Destruction Magic?
Staves don't train skills, nor is the damage they do affected by the appropriate skill, but higher skill level does allow for a lower charge cost when you use the staff. In other words, a character with a high destruction skill will get more uses out of a fully charged Staff of Fireballs than one who has a lower skill level
3.) Spellbooks - will more advanced spellbooks appear in a shopkeeper's/court mage's inventory the more you level up magic skills? Or are higher level spellbooks only found with specific NPCs?
The Spellbooks carried by merchants are based on your level in the specific school of magic as opposed to your overall level. General vendors don't carry Expert or higher level spells, so you have to get those at the College of Winterhold. Master spells are only available through quests.
If you're using Destruction, especially without mods, the Impact perk is a must. Though it seems kind of cheap, it enables you to stun-lock many oppenents- even Dragons! - and can mean the difference between life and death.
I found in my playthrough, especially at lower levels, that I did poorly when I didn't think like a mage. Until you get enchanting up to reduce spell cost, your magicka runs out quickly, so you have to pump a lot of points into it, leaving you vulnerable.
For a squishy mage, it helps to use the terrain to block enemies, use dungeon traps and gates against them and basically avoid going toe-to-toe. Shouts were an important part of my arsenal, especially Become Ethereal and Ice Form, which can both be used to aid a hasty retreat.
I retreated a lot.