Yes, Browning does manufacture rifles that can chamber the .17 Remington cartridge. Specifically, their X-Bolt series includes models that are compatible with this caliber. The .17 Remington is known for its accuracy and flat trajectory, making it popular among varmint hunters. Always check the latest product offerings, as manufacturers may update their lines.
yes
The smallest, best known centerfire handgun round is the .25 caliber (or 6.35mm). The practicality of the .25cal is questionable: you have all of the costs of a centerfire round, with not much more energy than a .22LR rimfire. In fact, nowadays, you are much more likely to find .22LR pistols than a .25cal. One step up from the .25cal is the .32 caliber. This is just about as out-dated as the .25cal, but there were some very popular weapons in .32 caliber, the Walther PPK (James Bond's weapon of choice) is said to be a .32cal (not a .380ACP). There are also pistols made to shoot rifle (long gun) rounds. These tend to be smaller caliber than a .25 cal, but are fairly modern (like a pistol designed to fire .17cal HMR).