In the US at least, the employer withholds based on your salary, unless you tell them otherwise, since that's all they know. If you have substantial income from other sources, you may need to tell them to adjust that number; there's a standard form for doing so.
Yes, gifts and bonuses are usually withheld from. They will rarely make a significant difference in your taxes other than that; remember that even if they push you into a different rate that's a different marginal rate and isn't likely to be enough to cause underwithholding penalties if you were already in the right general range.. Whether the company notices that and adjusts for it will probably vary from company to company; ask you HR department.