Is "safe with [someone/something]" a common expression?

Would it change the nuance of the context if I say "could" instead of "can" here?

In this particular context, you can use them interchangeably. There is a slight difference in nuance. "Can" expresses more certainty or a strong possibility, while "could" sounds more hypothetical and expresses weaker possibility, but the meaning is largely the same in general spoken English. Ex: She's not calling you? It could mean she doesn't want to talk. (The speaker isn't certain) Ex: She can be sick, that's probably why she's not calling you. (The speaker thinks this is likely)

What is the nuance of "here" being used at the end of this sentence?

The "here" in this video can be understood as meaning something similar to "[here] in this context/situation." Thus, we can understand "I know I'm not in the wrong here" to mean "I know I'm not in the wrong this time/in this situation/in this context." Ex: What's going on here? Why is it so noisy? Ex: I'm not at fault here.

What is the meaning of 'quite'?

'Quite' is an adverb meaning 'to some extent'. Similar words are 'fairly', 'kind of', 'enough', etc.

Do they mean the same? I mean "cycling uphill" and "cycling up hills"?

Yes, they basically mean the same thing here! "Uphill" refers to going in an ascending direction up a hill or slope, whereas "up hills" is meant literally - to ascend a hill. Thus, in this context they are interchangeable. Ex: I don't like walking uphill because it makes me sweat. = I don't like walking up hills because it makes me sweat.

Can I change "in eight hours" to "after eight hours"? Does the meaning stay the same?

Unfortunately, "after eight hours" by itself can't be used because it is not specific enough. The phrase "in eight hours" means "after eight hours from now". Only using "after eight hours" could refer to that eight-hour time frame beginning at any time. An example of how you could use the phrase "after eight hours" is, "After eight hours of sleep, I feel refreshed."