Hi John and welcome to the forum,
In regards to the most popular platforms, I don't know anything about Lazada so I can't offer any advice there. Of course, as you have correctly identified, the two big players in this space are Amazon and eBay. You can of most definitely use both platforms to sell from Phil, no issues there.
All you need do is open a Ph eBay account up Link hidden: Login to view
I think you will find this info interesting regarding Amazon Link hidden: Login to view
Any problems in regards to setting up an account, we'll be happy to help.
When it comes to suppliers, it really comes down to you doing your research so you can clearly identify everything you need to make sure you are giving yourself the best possible chance at success.
The Research Lab Tool results you are referring to are generated directly from eBay.com, they are very specific to that market. Now, that doesn't mean that they won't be successful in other markets, but it certainly doesn't guarantee it either.
It's only a tool, use it as nothing more as the research tool it is supposed to be. It gives you a good starting point to kick off your research and then that task can start taking you down all different avenues.
I hope that gives you enough to get you moving in the right direction but just give us a shout out any time you need any help with anything.