-If I buy a furnished house, can I interact with objects (like chests)?

Yes to some, no to others. You can interact with chairs (to sit), lights (to turn them on and off), books (to read them), crafting stations (to craft), and target skeletons (to hit them to test your DPS). There are a couple other miscellaneous interactables, like incense sticks that emit little plumes of smoke and personal Mundus stones occasionally available. Chests, however, are a no, and not for want of asking on the community's part.

--(If 'Yes') Can I store my stuff (or are the chests just for decoration)?
Just decoration. It's probably the most common thing on the community's collective wishlist.

-Can the exterior of a house be decorated (if said house has a courtyard)?
If your house has an exterior within the instance, yes, you can decorate it. This is true for most medium and larger houses, and about half the small houses. The best way to figure out whether a house has a decoratable courtyard is to enter the Preview instance and just look around. If you can get to a courtyard without going through an "Exit Preview" door, (i.e., by going through a different door) the exterior is yours. Note that not all houses that seem to have yards do have yards, though. The Silent Magnifico medium house is notorious for having a private roof that is outside the instance, and therefore undecorateable, so read where each door goes carefully as you move your reticle over it.

-Can I have crafting tables at my house?
Yep. You earn them through doing Crafting Writs. You can have basic crafting stations or attunable ones (which allow you to set the station to make a specific item set... great for crafted sets where the tables are out of the way or hard to get to)

-How do I bump up the limit of special items (Non-combat pets, etc...) that can be placed in my house?
Houses come in tiers, each with a set item limit (the bigger the house, to more furnishings and collectibles you can keep there). The only way to increase the limit at the moment is to subscribe to ESO+, which doubles all item caps.

--Is this based on the house's size? If so, does it show the limit?
Yes, though it doesn't show it before purchasing, which is a little annoying. This wiki page has the stats on all houses, including item caps: http://elderscrollsonline.wiki.fextralife.com/Player+Housing

Not really related to purchasing a house, though it is for decorating one;
-How do I make furniture? What level in Woodworking do I need to start making basic sets?
Level 1. To craft a furnishing, you need to find a plan (you can buy the super basic ones from certain NPCs, like woodworkers; the rest, you have to find in the world, like cooking recipes). Each plan will list what the crafting requirements are, and some will require multiple crafting skills on one character to make. However, the easiest plans start at crafting Level 1 of any given crafting skill (because there are furnishing plans for every crafting skill, including Provisioning, Alchemy, and Enchanting!). If you have the materials, the plan, and the skills, you can make it.

--Also, is there a limit to the amount of furniture one can have in their house?
Yep, all furnishings are capped, and overall increase is probably number two on the forum's collective wishlist. See the webpage I linked above for more info.

Hope that's enough to get you started!

Some resources for extra reading:

• https://eso.mmo-fashion.com/housing-guide/
• http://en.uesp.net/wiki/Online:Player_Housing
• http://elderscrollsonline.wiki.fextralife.com/Player+Housing