Your post is very confusing. You say that company A will not give you a written estimate. Then you refer to the "bid" that company A provided. First off, an estimate is an approximate valuation of the cost of an item, it is NOT A BID. A bid is a binding statement offering to perform specific work for a specific price, usually accompanied by a detailed description of the scope (what the company is proposing to do), the terms (payment, insurance, whether they will pull a permit, warranty, dispute resolution, and often many other items), and a time frame within which they will do the work. Bids are always in writing, as a verbal "bid" is essentially impossible to interpret.
You cannot compare two bids unless they both refer to the same scope of work. You need to write down the scope of work so any company wishing to perform the work will understand exactly what you want done, and any specific conditions you want them to meet. Specific conditions may include timeframe to perform the work, insurance they will be required to carry, who will pull the permit, when you will allow them to work, and any other conditions you wish to impose. If the companies you invite to bid do not like your conditions, they may elect to pass on the project, or they may suggest changes to the conditions, which of course you are free to accept or reject.
In the end, you would ideally like three or so written bids from reputable companies, in the form you require. You also need to specify the type of contract you want, this might include fixed price (all risk is on contractor), hourly rate (not recommended), hourly rate with a not to exceed price, cost plus profit, or one of several other types of contracts. If you are uncomfortable preparing a contract, you may want to consult with a paralegal firm, they can help you put together a simple contract that protects you from a wide range of problems that can develop on home improvement projects.