Setting expectations... If you assume you are going to get 100% of your payback immediately it's simply not going to happen... however I've gotten paid on ALL of the promotions I've fulfilled my end of the bargain on by using the "need help" button in the lower right corner of the History screen. I'd put my immediate payout success at around 60%, the rest get credited a few days later when NativeX gets around to looking at their messages.
The easiest, lowest risk/lowest reward options are the free "install and run" games or apps. Typically your payout is 1-5 shields and you need to, as stated, install and run the game or app. Read the text when you are reviewing the offers to make sure there are no in game purchase requirements, etc... Most (particularly the 1-2 shield payouts) will credit you for simply opening the app and then switching back to BoH, I will usually play through any basic tutorial since a few promos require this extra step but don't explicitly call it out in the offer text. A few games try to get you to link to your Facebook account (which I don't have) but not doing so has never prevented me from getting a payout.
Of the high payout "Free" promotions, the "FreeCreditScore.com" takes the cake at a 200 shield payout at 0 cost but you must call to cancel your account within 7 days to avoid a $10 monthly charge. I did this and it was no big deal, 5 minutes on the phone without any gimmicks or "please come back" stuff. The gold shield credit was immediate after I set up my account.
The Discover Card promo is also a big payout freebie (440 shields if you quality and don't mind opening and then closing a revolving charge account) but I did have to forward the "your card is on the way" email to NativeX to get it credited.
On the paid side, GoDaddy'com is a good value, register a domain for $8 and you get 240 shields (worth about $20 if purchased in game)... credit was immediate.
JibJab e-card is only worthwhile if you really want the service, $12 for a year of access for 74 shields vs 120 if you buy them directly for $10 in-game, but it's paid immediately.
Healthsource Multivitamin is a scam, period. The offer is a one month trial of some multivitamins for $5 shipping and handling for a 200 shield payout. I figured it was a good value ($5 only gets you 55 shields in game) and signed up figuring I'd call and cancel as soon as the product arrived, however the charge was denied and triggered a fraud alert on my Chase card... I talked to Chase about it and decided to leave the charge denied and just forget the whole thing... but a few days later a new $295.98 charge from "PPL*Shakeology 800-470-7870 CA" showed up on my statement. I called Chase and they indicated the "profile" (category and merchant info) was similar to the denied charge and had it reversed, then cancelled my card and issued me a new one to avoid further activity. Between the 30 minutes on the phone with Chase plus the time to change my auto-bill account on a number of sites I'll probably end up burning 3-4 hours cleaning up the mess. I've notified NativeX of the situation so they can follow-up with the vendor as well... irritating since I never did get the 200 shield payout for my trouble.

.. anyway, hope this information is of some use to other players, I'm sure there are other promos out there I've missed so feel free to reply/add to this thread if you have experiences to share on the topic.