I work at a mid-size software consultancy. In our region, the company's reputation is that of a trustworthy and flexible partner, with good engineers.
Our customers are mostly large companies (5k+ employees). It is not uncommon that, for a new project, we already start working for them before anything is on paper (more specifically, before an official purchase order was prepared by the customer). Middle management at the customers love this, because they usually hire consultants for urgent problems, but their internal bureaucracy prevents them from getting a purchase order out fast enough.
So, if your reputation is strong and customers will want to get back to you, I believe that point 2 becomes less important.
I once asked one of our managers why he didn't immediately get a particular verbal agreement on paper. He said "we have to work on trust anyway. We really don't want to get caught in a lawsuit, since it'll cost us a lot of money and energy - if there's a lawsuit, we already lost. So, we simply only work for customers that we trust".
The only case that I know of where we didn't get paid was when a customer went bankrupt. No amount of paper would've saved us there.
This is typically known as starting work "at-risk". You're right that it's a great way to show you're a good supplier. Usually starting work at-risk comes with the caveat that if the customer doesn't sign the contract and/or issue the PO in 2-3 weeks then work will stop.
You are very lucky to be able to pick and choose your customers and work with only those that your company trusts. Not all industries have that option open to them.
I have a friend, that was hired by a large media company to do some 3D models that the media company was hired to create to Siemens.
He never bothered in making all the proper papers, and just worked away.
Now Siemens got the 3D work, and ran away with the money... And how you can sue Siemens? They are so huge that if you attempt that they will make you go bankrupt in lawyer costs only.
In cases where this can't be done or isn't practical the next best thing is being able to provide contemporaneous notes of conversations, dates and times with people you have spoken to. I don't mean to be a little old lady "what is your name?" or anything like that.
I've used these many times in the past (not in court because it never got to that) in order to get something from a higher up by building a credible case and trail of evidence. This is known as "memorializing" that is you take and keep notes which you can use at a later date.
This is not simply "well I spoke to Fred J. and he told me that..." it's keeping complete track of back and forth with various individuals and interactions what they said and when. Many times someone higher up will simply judge whether the events could have happened [1] and the person on the other side will not have exact notes (why would they?). In any case it has worked for me.
And if it gets to court you have at least a leg to stand on, sound like you have your act together, and can present at least a cohesive case of your version of the events.
[1] Edit: Important that you don't exaggerate. The person higher up will judge the validity of the entire exchanges by whether they feel the lower level person would possibly promise or say what you claim. The more it sounds like what you said happens could have happened the more likely you are to get what you want in the end.
Not that I have had to put this to the legal test, but there is then this sort of written documentation, accessible to both parties, of what has been agreed and when. Having everything in an email folder is easier for me than keeping a log book.
As far as "legal test" things rarely make it to court (even with contracts). At the point that's happened you've already lost. They are normally settled before that for anything of a small nature. Anything larger (and it's hard to define small but I know it when I see it) needs a contract.
1) Never back down to a larger company. As soon as you give in and give them a break, you're going to be a doormat. Always be the squeaky wheel.
2) Always get everything in writing. They offer you a verbal deal? Say not until we both sign it and its on paper. Both companies and sales people will always balk at having something you can show a judge in a small claims court if its just an empty promise. Then you have something to hold their feet to the fire if it doesn't work out.
3) Always, always, always be willing to walk away from a deal. It's a big account? You could save yourself hours of frustration and use that energy to find several smaller companies. Or you could fine another larger company who will treat you better. Nothing is worth being someone else's patsy, nothing.