The first question I ask clients who are looking to be paid in stock (whether they are founders, early employees, or contractors) is what are the terms of the stock? For instance (I am assuming here that you are a contractor rather than and employee), it is likely that (if they are setting up wisely) the startup's "founder's stock" is subject tot a vesting schedule. If you receive "founder's stock", you will be subject to the same vesting schedule. Ask yourself if that is realistic...if your work for them will be completed in six months, but the stock you receive is on a three-year vesting schedule, you might want to re-think accepting stock on the same terms as the founders. That being said, assuming you are ok with the stoic you are receiving, here are answers to your questions:
1. A "stock grant" to a contractor and receiving stock as payment for your services are pretty much the same thing. Is there a different understanding of the terminology that is troubling to you? Bottom line is that, as a contractor, you aren't receiving the stock as a "gift" or an incentive payment (like an employee would). (In some states, corporations aren't even allowed to "gift" stock - if the corporate issues stock it has to be in return for consideration (i.e. something of value).) No matter what you call it, you are receiving it as direct compensation for your work. You will need an agreement that makes clear that the stock is issued as compensation. Your basis in it for tax purposes will be the value assigned to the stock when it is granted to you or when it vests (unless you file a timely 83(b) election with the IRS - which calculates basis at the time of grant regardless of vesting).
2. If the stock is paid to your company, it will be owned by your company. So, you will not, as separate individuals, be able to make different decisions regarding "your" shares of the stock. The company will be the one to benefit directly from any appreciation in the value of the stock. And, if you want to distribute that value to you and your colleague, you'll have to pay the various levels of taxes applicable when it runs through a C-corp (i.e. corporate level tax and then personal level tax). If it is granted to you as individuals, then you each have control over when/if you dispose of it and the proceeds come straight to you and are taxed at the personal level only. Keep in mind that you or your company will also be taxed on the value of the stock granted as income (1099-MISC).

Happy to chat further.