Yes. To both instances.
Commercial licenses allow you to use the font commercially. In other words, you are allowed to make money off of designs which use commercially licensed fonts. If you are charging anyone anything for the items you are creating with the fonts, you need a commercial license.
The license is granted to the individual or company purchasing the license. You are allowed to use that font as often as you want in any project you want without repurchasing, provided you remain with in any "seat" restrictions (1 computer = 1 seat some licenses have seat limitations). However, You are not allowed to share or give the font away for any reason.
If your client wants the font - as in they want to be able to create their own items in that typeface - they need to purchase it as well (or you purchase it on their behalf). The client does not need to purchase a font simply because you sent them a .png image which uses the font in it. Or because their vector logo has outlined characters from the font that have been modified. They only need a license if they want the font in a font file format.
It may be beneficial to actually read the license you get from each foundry you purchase from. Yes they are boring and tough to get through. But it may educate you considerably.