Hi all. Happy to join you here! I'm a freelance copywriter just starting out. Are there any web designers who could give me a little advice?

Question

I will be cold pitching web and graphic designers in the next few months (sorry!) but  I don't actually fully understand the nature of your relationships with copywriters. Can anyone answer a few short questions for me please? 
Hopefully one day I can repay the favour :)

1. Do you work with a copywriter?

2. Do your clients ever ask you to recommend copywriters or do they send their own copy?

Lastly,
3. I was going to email offering a free piece of work (ie. a polish or rework of stuff on your site) to show what I can do and in a shameless attempt to curry favour.
-Do you get many cold emails from copywriters?
-Is this an approach you have seen before/ what is the usual approach?
-Would this offer be attractive to you if you received it?
-What do you need most from your relationship with a copywriter/ what would actually be useful for you that I could offer?

Many thanks in advance for anyone that can help,

All the best,

Pete