Before I say anything else, thanks for providing the link in your signature ... your artwork is beautiful. I pride myself in my drawing abilities, but I am nowhere near in your league. It seems to me you've made the right decision to pursue what is obviously a passion for you, the world will be a better place for it.
On the aging delay ... 14 months is a very long time. There is no telling for certain whether you are still in affected by the aging delay, but the symptoms you describe sure sound like it. Some sites seem to emerge from the delay w/in 6 months, mine took over 12, and I've heard of a few that take up to 15 or so months. Unfortunately, other than doing things right, there's nothing you can do but wait if that is the problem. When it happens, you'll likely know it as some of your pages jump up closer to the top in Google. But you do need to be aggressive in promoting your site, even if you're not able to compete yet due to the delay. I'd definitely get away from any bulk link exchanges as mentioned before.
When people speak of keyword research, they mean being more aggressive than just guessing what you think relates to your site. Use a tool like Keyword Discovery
to do some actual research. Wordtracker offers 24 hours of access for $8 US if your budget is strained, I did just that my first visit. Make a list of every phrase you think people might use to search for you, then run those phrases through one of these tools. They will give you data on what phrases people are actually searching for that directly relate to the phrases you input. If you see that certain phrases are being searched for regularly, then write up some content for your site that includes those specific phrases. It's really pretty simple and straight-forward, You need to be using very similar language on your website that people are using to search in order to rank for that phrase when they do search.
When people speak of quality links, they are generally referring to sites that have strong content connection to what you offer, or are related in some way to the same customer-base, and ideally sites that have a lot of traffic by people who would be interested in your work. Look for hubs or authority sites that pet owners rely on for information. There have to be a lot of sites dedicated to each breed of dog, each breed of cat, mixed breeds, pet care and feeding, pet grooming, pet memorials (funerals), etc. If you have to exchange a link then do so, but at least you are targeting quality sites that are related to you with your link requests.
Most of your link requests will go unanswered however, even with good sites that would be a natural fit. That's because the webmasters of these quality sites get an overwhelming amount of requests. So be creative. Your site seems like a great candidate for a postcard campaign as you have beautiful artwork to feature. There are services that will do the printing and mailing for you if you provide a list.
What Scottie was referring to is that a successful web-based business must have a broader marketing strategy than strictly search engine traffic in order to be successful because search engine results are volatile. You need to investigate different ways to advertise your service. Reconsider adwords - though it can be expensive until you get it right. A 'successful' adwords or PPC campaign will pay for itself, but it takes a lot of effort to learn to do it well. Consider advertising directly on some of the authority sites mentioned above, they cater to the very people you want to reach. Also consider setting up an affiliate program - they are slow to build and take effort, but a couple of good affiliates could send you a lot of very qualified traffic. The idea is to use all of the advertising venues available, because google traffic could stop overnight, and if you rely on it too heavily, that could put you out of business.