There's a fine line between persistence and simply being an annoyance, however I've seen some great examples of people balancing these properly.

Usually this comes with the ability to be very human and personal. One way I've seen this done is with old school hand written letters. On a few occasions over the past few years I've had a vendor reach out to me with a hand written letter. You just don't see them anymore, so they really stand out.

Within the context of the letter I'd consider making the ask for "feedback", not for the work itself. I would explain your situation very personally around how and why you are seeking feedback, and also make it very clear that you respect their time and understand that no answer is OK too.

There isn't a 100% solution that always works. But making it short and personal could be a nice touch.

Twenty plus years ago I would call tv correspondents and producers at local news stations to ask if they needed an intern. Many of them said yes when can we meet? I finished college with nine internships in print, tv and radio (one at ABC News in NYC) all on my own accord and persistence. I graduated with multiple job prospects at major networks In NYC. I got those opps by being persistent and savvy. Knowing when to call decision makers before or after their assistants would be away from their desks. Or my classic move was a four week blitz of sending my CV to the same person leading to many interviews.

I have an unbelievable career because I have been persistent and take risks. If you have the courage to do what needs to be done to advance your career or business you will be fine. The people you approach that want to punch you in the nose as you say are not the type of people you want to deal in business with anyway. I'm here if you need more insights.

It is perfectly appropriate.

Persistence is necessary in the sales process. You need to be polite and respectful of people's time, but multiple contacts is usually necessary to close business.

People have an automatic filter against being "sold" something, and often times in order to get through that filter it takes multiple attempts. Most of my clients told me no the first time I approached them, but diligent follow-up had lead to closed deals.