You can use Moz, WooRank or other tools to review a site, obtain a grade along with other helpful information; however, you cannot depend on these tools to even come close to performing the type of analysis you are seeking. To use an analogy, you can use a template to draw up a will or other legal document, but if it is really important you would always work directly with a lawyer. The same applies for other fields whether it is a medical opinion or SEO advice.
Let's break down one component of your request: "URLs ok? :: Is the URL mapping ok, old to new, best web page?"
What EXACTLY is meant by URLs ok? A full breakdown would be:
1. Protocol. Are pages served as http or https? Which is best for the site?
2. Is the domain name optimal for the site?
3. What is the best URL paths for the site? If you operate a clothing e-commerce site should the primary category be the brand? gender (men vs women)? kids? something else such as clothing type (shirts / pants / suits / etc). That decision cannot be made by any tool.
4. What about id numbers? For a small site, a url will appear cleanest without product ids. For a large site, an id number is needed in the URL to ensure fast page loads as each URL needs to be delivered by a database and text searches take too much time. Even if you have a small site now, what are the future growth plans? How does a tool adjust for these factors?
5. Should the brand name be included in the URL? How about the product name? How about the product description?
6. Best web page? Sites often provide numerous pages on closely related topics. Let's say you asked on Moz what is the best page to discuss web crawling tools. There are numerous articles discussing Moz's tool, Zenu, Screaming Frog and other tools. There are articles which compare web crawlers. How can a tool look at your URL and decide which page is ideal? By date? Text match? Google has billions invested in trying to provide such feedback and they clearly get it wrong at times.
There is certainly feedback a tool can offer. For example, if a url ends in .html the tool can recognize the technology extension and recommend to drop it. Very simple. With enough time and money, tools can be built to do almost anything. You presented a laundry list of needs, each of which requires a professional opinion. Even if you were to find a tool which provided responses to all of the inquiries you shared, I am highly confident the responses would not be optimal.
I hope you find this response helpful.
-Ryan