Welcome to teh escapist!
Thank you for not playing on Adept and then whining because easy mode is too easy lol.
To answer your questions all at once: of course you can! This is
The Elder Scrolls! You can have it your way, man!
If you have trouble against Magic, play as a Breton. You will take less damage from all spells. In addition, the Dragonskin power grants you a 50% chance to Absorb spells for 60 seconds once per day. You cancel out any spell's damage, adding it's power to your own Magicka pool instead. You can make that Absorption permanent regardless of race by finding the Atronach Stone. Receiving it's blessing will raise your maximum magicka considerably, decrease your regeneration rate, and grant permanent 50% spell Absorption. These are powerful abilities for all types of characters, even (sometimes especially) those who do not use the magicka they gain by using Dragonskin or the Atronach stone.
Keep a lot of "Resist X" potions around. They are easy to find or purchase. Don't worry about the cost, you'll be rich eventually one way or another.
There's not much to say about playing without magic except that it's very viable. Just keep in mind the above. There is "Resist x" equipment to be found if you do not wish to craft your own. You may find keeping a bow around makes your life easier, but you can complete the game without ranged attacks. Also, you can make up for the loss of magic with scrolls and especially shouts. Don't forget your shouts!
It is very possible to play using only magic. You may find yourself vulnerable if you run out of Magicka, so don't sell all your potions. Try to acquire some gear to boost your regen rate. The College of Winterhold will provide you with some. Also, Staves serve nicely in that situation. They're effectively just spells that consume soul gems up front instead of magicka upon use. I recommend the school of Alteration for the Magic Resistance perks. They're exactly what they sound like. Breton + Atronach Stone + Magic Resistance = fuck you, Novice Necromancer.
Just be aware that playing pure magic-only is difficult on Expert for the first few levels because your magicka pool is not large enough. It's not that hard to overcome, really, but it might leave you with the impression that magic is too weak. Dump all your level-ups into Magicka until you have enough to start balancing it with some Health. Playing as an Altmer (High Elf) eliminates this problem because they start with extra magicka and a 60 second regeneration boost they can use once per day.
Planning your perks helps.
Use this to plan and save your builds. Personally I like to pick about four skills and max them out. But sometimes I skip a branch or take just a few from a particular tree. Building your character to suit your preferences is the whole idea.