I'm also getting this problem with Portal 2, even though it's a paid game. Luckily I completed the Vampire Masquerade quest, but not Apex and I don't know if I'll be able to complete the Portal 2 and Dota 2 quests.
For me, after some struggling, it finally counted 50% progression for Apex and Vampire. But after a while when I paused the games, and wanted to resume, both progression bars went back to zero again.
Plus, Dota 2 cant be added to Steam Library. Its probably a Steam thing. But I was not successful to add the Dota 2 free game license to my library. I googled and many people seem to have the problem of "Dota 2 not in steam library"
For Dota 2, you need to play for like 30 mins to an hour before it'll register to your account I believe, then you can do the quest. Works the same with TF2. Means you have to play more the first time around, but yah.