I'd rather look at it from this perspective: create a different AdWords account for a different business.
For the rest, Lee is mentioning great points.

This strongly depends on what product you're targeting, what market you're in, and countries you're after. Also, is it an online product or a physical product? How long are you running the ads for? Usually a good 2 weeks is fine, then the results decline. It would be great to know a bit more details about what exactly you've done and then figure out with you where exactly the problem is.

I love this question!

As a former Googler and Digital Marketing Specialist with my own company, I hope I can offer you some insights.

Finding clients is no issue for me and I'm luckily in a situation of having more demand than I can handle. I love it this way because I can much better plan the business and really dedicate my time to work with clients I consider a good fit and are really ambitious to grow.
The most important aspect to focus on here for growth is to deliver highest quality work and dedicate to 110% excellence. Make exceeding your customer's expectation you highest priority. Everything else will follow.

Wanting to hear from digital marketing agency gurus out there. Folks who have started and run successful digital agencies.

If you could coach your younger self and your younger self was starting an agency from scratch and gaining clients was NOT an issue, what would say to focus on to grow the business as fast as possible, but grow it well?

In other words, what are the pitfalls someone without your experience should avoid and what are the shortcuts someone with your expertise would take to grow fast, but still manage clients and provide a great product and great service?

1. Who should be the first real hire?
--> A virtual assistant. You can't do everything your own and this person will be crucial in freeing you of time-consuming tasks that add little value to your work or clients.

2. Are there any aspects of digital marketing that should be either avoided altogether or should only be added once a good foundation is set? (i.e. web design/development)
Focus on one area in the beginning: e.g. Google AdWords/Analytics setup/optimization/management. Don't do all at once. I see a lot of agencies offering a ton of products but not really doing them well because of a lack of focus. Once you have a solid business and customer base, I'd advise you to start of with partnerships and collaborations for other areas of digital marketing. If that goes well, then consider hiring a person in this specific area.

3. What is the best but fair way to secure recurring income while providing value worthy of it?
Depends a lot on what you offer. However, if you offer high quality work, you will always have customers running to you. Monthly management is definitely worth it, but make sure you do it well and really offer concrete action steps and not just a look at the account once a month.

4. Would you start with outsourcing to India or the Phillipines et al. and if so, what positions are best outsourced to these locations?
No, I hired my first employee from Kenya. India and Phillippines are stereotypes and it is very hard to find someone that understands you, can communicate well, has a good level of English, and is reliable. There is 200 countries in the world - look outside of the typical suspects and you will find very valuable gems and talent.
Positions to be outsourced: can be pretty much everything, from a VA, to a webdesign VA, account manager VA, etc.
Also think about your company's focus - if your focus is also outside the US, you might want to invest in hiring a person part-time in your potential target market.

5. What are other ways to make sure clients feel taken care of even if you may not yet have the full staff just yet to manage everything on point?
Communication. Communication. Communication. Super important. Plus, if something goes wrong or not as expected, tell them. Nobody expects you to be perfect but to find a solution to a challenge/problem.
Create your own list of clients and a system - how often do you catch up with them? Make notes of personal preferences or hobbies - include small talk and show interest in their personality.

This answer got now much longer than I even expected but I hope it helps and offers valuable insights.
Feel free to set up a call - I'd be happy if picking my brain supports your growth and expertise. ;)