I had no idea this was so confusing, and I think what is hanging us up here is the language and usage. Patrick's answer is correct, but I think it *still* could be misunderstood because of the way he repeated back some of your statements.
In reply to:
1) We register with DH and even if we bring our own domain name, DH gives us one.
Yes. Same answer as Patrick gave. A little clarification might be that they may give you one, or two, or more, depending upon the promo code you used. *Everybody* gets 1, but some who used promocodes that offered more may have additional "free" domains as part of their "plan"
In reply to:
2) If we use the freebie we now have 2.
Yes ... two domains, that is (one of which was free). If you use the free domain(s) that are part of your plan you will have a total of the number of domains you already own (registered anywhere at all, including DreamHost) plus the additional "free" one(s) that came with your plan.
In reply to:
3) Next year, still with DH, we get another = 3.
No, not for "free" you don't. If you have not used the free one(s) you had as part of your plan, you can still add them, but they don't "add up". You will always have, at any given time, the total of whatever domains you have already registered and renewed, and the additional domains that came with your plan for "free", but they do not accumulate.
In reply to:
4) It's time to renew with the registrar. My original is GoDaddy. I pay them for domain 1 but DH covers 2 and 3 for as long as I stay hosted with them.
Nope. See question 3 above - you will still have whatever number of free ones came with your original plan.
In reply to:
5) They just told me recently that they still owe me a domain but they didn't tell me that I can get one every year. Hmph.
Because you can't. You can only get another "free" one if you let the first free one expire (or any other combination you want, as long as you pay for all that you have other than the original number than came with your plan). You *do* get another free year of renewal on that "free" domain, and that continues as long as you have the plan or the domain - if you do not renew the domain *then* you are back to having another "free" one available.
In reply to:
I'm past one calendar year from my initial purchase so I can get one for this year and now I need to wait about 10 months to get another.
Nope. You can get one whenever you want, and that will be your "free" one. If your plan comes with a single "free" domain (as most do) than that is all you will get, though you can keep that one and it will remain "free" (no renewal fees) for as long as you keep your hosting plan account with DreamHost.
Of course, you may not want to do that, as you may want a *different* domain, and no longer want the original "free" one you chose .. that's fine, you can just let it expire, and then you will be back to having "0" free domains on your pland, which comes with 1, so you can *then* add another.
Obviously, this could get confusing, so DreamHost just credits you with the amount of one of your registered domains each year .. that leaves you to sort out "which one is 'free' " in which ever way suits you. 
--rlparker