Web designers back link on client sites

The alt attribute is not valid to use with the <a> tag so you can remove that completely.

You only need to put the title attribute if you need to explain further what the link is for. That's probably not necessary in your case since I am sure that link will be self explanatory (I am assuming it is preceded by wording similar to "Designed By"). If you use it, it should not be the same text as the anchor text. That's unnecessary and unhelpful.

Using the same content in the link, which is probably your company name, is fine and won't be problematic. It also helps to make sure you rank #1 for your company which is always a plus. You may want to consider expanding the anchor text to include keywords like "website design by" or "website developed by" to help you rank better for search terms containing those phrases or something similar. However the competition for those phrases is so high that it isn't realistic to expect to rank well for those terms in this lifetime.

update

Yes.