2. Lever, bolt, or semi-auto? Let's rule out bolt-action for home defense, as follow-up shots and ammo capacity are lacking compared to lever guns and ARs. So between lever guns and ARs, which to choose? This is a tougher choice than it first appears. ARs are great - powerful, with fast shots and virtually unlimited ammo capacity, but also the most excoriated gun in America and the most likely to fall under restrictions at local, state or federal levels. Bring an AR with a flash suppressor or a bayonet mount into Connecticut and you're liable to face felony charges... but out here in AZ, ARs in any style or denomination are as common as SUVs. Lever guns, probably because they are the quintessential American deer rifle, will likely be the last repeating rifle to be banned. They also have a pretty simple "manual of arms" - you don't need to know you only load 28 rounds in a 30-round mag, you don't have a forward assist, and you won't get confused hitting the mag release when you meant to hit the bolt release. So don't rule them out.
2a. Calibers. In the AR, the basic caliber is the 5.56x45/.223. There are lots of other chamberings, from .204 Ruger to .458 SOCOM, but the 5.56 is the most prolific and the easiest to get ammo for. Lever guns also have a wide range of chamberings, but the most common today are deer rounds (.30-30 and .35) and the pistol rounds (.38/.357, .44 and .45) which are commonly used for Cowboy Action shooting. For home defense, assuming your home isn't the Southfork Ranch, the pistol chamberings have a lot going for them in lever guns. Reduced recoil and muzzle blast and greater capacity over the rifle rounds are the big advantages. The extra barrel length of a rifle over a pistol adds extra wallop to the handgun rounds at the receiving end, so for home defense (HD) a lever gun with a dozen rounds of .357 or .44 Special is pretty formidable.
3. The basic Marlin 336 (.30-30 or .35) is around $500 new these days, but there are lots available on the used market for lots less - check GunBroker for examples. The pistol caliber guns are pretty popular so they'll run a bit more, either new or used. A good red dot like the Aimpoint H-1 will run you $400 or more, plus the cost of the mount, but if you're willing to take a chance on lesser products the Bushnell TRS-1 is about $100 and Tasco, Barska and others also have low-cost offerings. Just remember that you usually get what you pay for! But a red dot sight on a pistol caliber lever gun is a pretty cool HD tool... nothing to sneeze at!