GH: Yes, but that is in control of the user. For example, if you visit a site from your iPhone and tap the "plus" button in the lower browser navigation, you'll have the option to "add to home screen," which will put an icon on your screen similar to an app. Here is a good tutorial.
This shows you how to customize your icon so that if others do this, they get the icon you'd like them to receive.
3. How does one go about developing a budget for the mobile site project?
GH: Eighty percent of your marketing budget. Just kidding!
This really depends on what type of business you have. The budgeting process shouldn't change just because it's a mobile site. I'd ask, "How did you budget for your current desktop site? How did you budget for email services or direct mail?"
You have to determine which solution is best for your business or your client's business considering execution, management and price. Some of the DIY template-based solutions are only a couple hundred bucks to get started.
If you need something more than that, you might need to hire someone or an agency to help.
4. How do you recommend testing the effectiveness of offering a responsive site vs. not offering a responsive site? We often need hard numbers to convince clients to invest/change.
GH: Unfortunately, I don't believe there is a concrete answer for this. The best way to know the effectiveness of a responsive site vs. not is building one and going through the process. You need to understand the challenges, time commitment, efficiencies, etc., that result from that experience in order to compare it.
Most clients won't come to you saying, "I want a responsive site." It's your responsibility to recommend the best solution for them for as long-term as possible.