It should show up as 1.14.3-Forge 27.0.??? in the profiles. Also unlike with Forge 1.12.2 and prior it shows up among the 1.14 versions not below Release 1.0 where all the prior versions of Forge show up.
I'd suggest getting the offline installer since you locally have it and can just execute it (if you have a 64bit system then download the 64bit version if you have 32bit uses the offline version above the 64bit one).
If those don't work I'm not too sure, but if they do then your ready to install mods, which you can do from here safely and not from another site that is typically dodgey, has malware in the files and so on (ignore Fabric mods as they don't support Forge they support only Fabric, if you want to know about Fabric check my signature and the thread for anything you need to know about it from a community member for the community ):
I have no idea then, since if you have all that is required, and know the forge version to select in the installations tab then once you have the mods, core mods/libraries/apis and can work out any conflicts with them your game should work.
Have you watched a tutorial at all on how to install it to your launcher?
Otherwise the only other thing I can think of is if it's a hacked launcher or something else and in that case I can't help you with that.
You'd have to show me your launcher in a screenshot since I don't know what else to suggest without one. And I'd usually recommend another launcher but I don't think Forge works in MultiMC yet (don't remember if it is supported yet, only Fabric for 1.14 versions. Edit: just checked and you can't easily with it yet by just the button), but Twitch you could setup if you wanted but then again that can be awkward to setup too sometimes.