No matter which skills you level, you get the same attribute choice: 10 magicka, 10 health or 10 stamina + 5 carry capacity.
If you only level (for example) Alchemy, you'll get some levels and the attribute and perk points that come with them. Without other skills, you might be very restricted in how you may spend those perks - or you can save them until you get enough skill to buy the perk you want. Without any combat skills, the world in general might be a scary place - however there are still low level bandits for you to kick around to get combat skills.
Absolutely.
Skills
Illusion synergises well with sneak as invisibility does not gaurantee not being detected. The CC spells are also very useful, allowing you to control situations that would otherwise overwhelm your squishiness. Silent casting will be vital, so get it asap.
Destruction will be tricky without deep magicka reserves until you start to enchant your own stuff to reduce cost. But Impact will work beautifully for single target assassinations. With silent casting, I believe you could also use runes for some fun, though on master they will quickly lose effectiveness. I would stick to one element to save perk points (perhaps shock as a useful rarely-resisted element), and don't bother with master level perk. This will be a useful tree against dragons.
Restoration is moderately useful with the respite perk, allowing you - in conjunction with light armour stamina regen bonus - to spam dual wield power attacks all day long. I would not worry about the perks above adapt. Get respite, the magicka regeneration perks, and possibly the "auto heal on zero death" perk that can help out if you are smacked around when dual-wielding.
Light armour is great, and will give you survivability that most mages don't have. It also takes up far fewer perks than Alteration.
I would skip conjuration perks completely, unless you wish to use bound weapons as a role play. You may ultimately wish to level it to 90 at some point in order to provide a non-smithing source of Daedric weapons (see smithing below), and if so don't rush on it until you need to level other skills up to level in general (high 30s onwards). Alteration has few uses in this build.
Enchanting will be useful, allowing you to supplement your lower magicka with regen and reduce costs. Don't bother with -100% cost on any given skill tree, as you will probably only use destruction as a filler against mobs and dragons but if you use it a lot against high level single targets then consider -100%, as the Expert level spells are expensive. I would aim for some +magicka regen (or +magicka - calculate which is more effective given your base magicka) as that can be used for all your magick trees, and you will be using them all not just 95% destruction like in pure mage. Possibly some reduce cost for illusion. All other slots for boosting your melee and resistances. Restoration is fine without reduced cost.
Smithing is great - but you may not feel like role playing it. Depends on your target equipment, but you could make do with steel smithing and the enchanted item smithing perks. You can get daedric weapons through the Mages guild, so don't feel obliged to perk up smithing.
Alchemy complements both skill sets well.
The Beginning
Don't spread yourself too thin to start off with. I would work on illusion, sneak and one-handed until you can comfortably stay hidden in shadows and get silent casting perk. From there you can start on destruction up to Impact, to have a useful fall-back of dual-casting firebolts (it can interrupt dragons and nasty bosses), and get restoration to apprentice level. I would then work on getting illusion to 75 and sneak till 60 for backstab bonuses. By this point you should be able to flit in and out backstabbing and skirmishing, using Invisibility as a means of getting back into cover.
Quests
You will not be locked out of any, and can complete Thieves guild, DB and Mages guild. They all work well together. Thieves guild and DB synergise well, and Mages guild unlocks some tasty equipment.
From here, you can choose. I would work on your sneaky sneaky and melee skills, as you will already have the main utility from your skill sets.
Use illusion to ensure you don't get hammered too much, and don't be afraid to go invis and hide. Your main advantages over a pure stealth character are your much better flexibility when detected - so use these advantages.
Hth =)
EDIT: stats: ymmv but I have had good sucess with balanced melee/mage with 2:1:1 health:stamina:magicka. I leave stamina at 150 (maybe 200 if going dual wield).
EDIT2: added link to find out if +magicka or +magicka regen, as per the paragraph on Enchanting. added link for Mage's guild and daedric weapons
Best Answer
Don't worry, your loot will be safe.
Hearthfire's new houses aren't actually in place of any of the houses that were there in vanilla Skyrim - instead, three areas of the countryside are converted into potential building sites for Homesteads.
Your purchased houses and their contents, therefore, won't be altered.