Chirality on Propranolol
As stated in user208322's answer, $\ce{CH3}$ and $\ce{H3C}$ are exactly the same, they're only written differently for ease of reading. As per your question about how much truth there is to having 4 groups, it's 100% truth. You cannot have a chiral carbon that doesn't have 4 groups attached to it (i.e. must be sp3 hybridized). If you think you've found a chiral carbon that only has 3 groups attached to it, you're probably forgetting the unwritten hydrogen.

In this example specifically, there is only one chiral carbon, the one attached to the secondary alcohol. Although it may look like the carbon is attached to the same group twice, you need to look further than just the first atom. Yes it is attached to two carbons, but one of those carbons is attached to a nitrogen and the other is attached to an oxygen, so the groups are different. Two groups are  only the same if you can follow it down every single atom and they match exactly.

Edit: In regards to your question about following the chain all the way down, realistically you only need to follow it until you see that it differs from the other chains. So for chains 1 and 3 once you get to the nitrogen and oxygen atoms you see that they're different and there's no point continuing. And the unwritten hydrogen has to do with the fact that organic chemists use shorthand in their molecular structures. Basically, carbons and hydrogens are extremely common in organic molecules so they just don't write them. All the points are carbons unless otherwise labelled, and they must have 4 bonds, so where you see that the carbon is only bonded to 3 things, it is also bonded to a hydrogen as well. See this wikipedia article for more info (specifically, this section).