1. Cast bullets can be used in both handguns and rifles for a broad range of performance, from plinking ammunition to hunting pretty large game animals. The higher velocities and pressures require the best possible fit of the bullet to the lands and grooves, adequate bullet alloys (hardness) to withstand the pressures, and sufficient bullet lubrication.
2. Gas checks are typically a copper alloy cup that seats on the bullet base (requiring a cast bullet intended to accept the gas check). The purposes are to protect the bullet base from flame temperatures of the propellant powder, and to provide a scraping action through the bore to minimize leading transfer.
3. "Hard cast" generally refers to the use of bullet alloys containing metals that are harder than pure lead, such as tin and antimony. The higher the levels of alloying metals the harder the finished bullet will be. Harder bullets are less subject to deformation under pressures of firing and passing through the bore, and exhibit less deformation upon impact. Very hard bullet alloys can become brittle to a point that impact may result in fragmentation.
The above is a real thumb-nail explanation. I regularly use cast bullets in handgun and rifle cartridges ranging from .25-20 Winchester up to .45-90 Winchester and .45 Sharps Express, and at velocities from less than 900FPS to over 2500FPS (depending on cartridge and application). I have taken hundreds of different small game animals and birds as well as deer, antelope, and Rocky Mountain Elk (live weight over 600 lbs.). I generally use bullet metals of about BHN 12 for handgun and BHN 16-22 for rifle cartridges. BHN means Brinell Hardness Number, an industry standard for rating relative hardness of certain metals such as lead alloys. I do not use gas checks for most handgun loads, but I use them for rifle loads (.25, .30, .32, .33, .35, and .45 calibers) at velocities from 1400 to 2500FPS or so.
For an excellent primer on cast bullets in general I would recommend you obtain the Lyman Reloading Handbook, and if possible a copy of the excellent NRA publication "Cast Bullets" by Col. E.H. Harrison (long out of print, but occasionally seen on eBay and other on-line sources). Everything from bullet alloys, gas checks, bullet casting, sizing, lubrication, and specific advice for just about any application, presented for beginners to advanced users.
I've been casting bullets since 1973. It has been a long, long time since I purchased any factory ammo to speak of, and I find that cast bullets meet nearly all of my shooting and hunting needs. Also one of the most fascinating aspects of the shooting sports.
Best regards.
