Is there such a thing as a first right-of-refusal for an equity stake or C-level position?

This sounds incredibly convoluted and problematic for you. Generally, any financing structure that is this unique is a turn-off to new investors, especially institutional investors. That they've recently raised funding for their business but are contemplating building a new business with you (as a subsidiary) is a concern unless this was part of the plan they articulated to their investors and investors are aware and supportive of this initiative. Finally, it seems as though you are being asked to invest a lot of energy in something that is entirely dependent on future funding validating the business, which is always a significant risk.

Obviously, I'm missing a lot of details and so happy to talk in a call but I would say based purely on how you have defined this situation, that it seems more trouble than it's worth to you.

I'm not an attorney but if this was my company I would probably put some kind of contingency clause in your vesting schedule that makes vesting of shares contingent on your performance and the funding being raised.

You can document such an agreement in any way you like. Typically, you might see provisions regarding contingent employment & incentives in your consulting agreement. For an agreement like this to be valuable to you, you would want to set forth the terms in detail, including a good faith effort to obtain Series B, the terms on which the executive position/equity is offered (i.e. as a ROFR, automatic trigger upon financing, performance-based or other metrics, etc.), what exact position/responsibilities would be offered, what type and how much equity would be offered and on what vesting schedule (or, if preferred, you can leave these terms vague). I think it would be unlikely that a startup would want to get that specific in advance. However, it can be tough to put agreements of this sort in any sort of binding writing because the needs of a startup can change over time depending on performance, vision, investors, etc. It's a double edged sword: If you don't put these agreements in writing, then they are not binding, but if they are in writing, they may end up being out of line with actual need of the company (or you) when that time comes along.

So, don't be surprised if a startup doesn't want to be bound to anything in writing and make sure that your agreement for consulting services is fair without assuming you'll get future employment or equity.