This falls in with learning the basics in design and knowing your market. Some questions you've asked should really be implemented with a vector. I would strongly know the market you are trying to design for and learn the workflow.
Typical resolution format is 300dpi. Photoshop can be used but should be used with minimal cases and not relied upon. I would suggest mainly using Illustrator or InDesign, vector based. Since you would be designing it in vector it can be re-sized and resolution is not an issue.
Now there are other standards we tend to fall too. If its a billboard design and won't be scene for large distances theres no reason to print @ 300 dpi and its knocked down to 150 dpi.
Default resolution is known as 72 dpi and even if you design at higher resolutions the browser will still only display what pixels exist and this has been discussed here. I typically do my wireframes @ the 72 dpi level but lately I have been going to straight code because designing for web is starting to become more of a problem. i say problem because what can be rendered as code is usually off to what you designed and clients are noticing and comment on this. I see and understand why some design at higher dpis for great detail but this is an area that is constantly evolving.
When I refer to learn the basics I also refer to the KISS effect. If you are going to do a full branding, such as business cards, web, etc., etc., a vector based logo can still be applied across the board. If you design in a pixel application you will spend triple time working on the resolution/quality of the artwork to look good when applied to the medium.