It is not strictly necessary to use a specific time reference with the Past Simple.
It is the other way around: when you use the Present Perfect, you better not use a specific past time reference:
I have watched this movie yesterday. (invalid sentence: the time period mentioned - "yesterday" - has already ended)
Concerning your question,
I just watched the first episode.
Is an appropriatetly formed sentence in the American English. British English speakers prefer to use the Present Perfect:
I've just watched the first episode.
Reference
- Comparison of American and British English - Wikipedia
- An answer to a related question at ELU.
You should use the appropriate tense at each point. You start talking about the summer in the past tense. No problem here.
This summer was horrible, I didn't even get to go anywhere.
Next you change to the present perfect. That's not an error. But the use of the word struggling is awkward, it looks like translationese. Better to use past tense for the accident and present perfect for your current state:
My brother had an accident, and we've been helping him recover.
"B" can also use whatever tense is appropriate. First sentence is about the past, so past tense. Second sentence is about the present, so present tense. The "I'm loving" is casual (continuous tense of stative verb), but I think acceptable here
We did loads of things and I had lots of fun. Like now, for example, I'm in New York! I'm loving everything about it.
A sentence can have mixed tense:
Yesterday I went shopping, as I will go to a party tomorrow and the bag that I had bought to take with me was too small.
Each finite clause has its own tense, and while jumping around tenses too much is confusing for the reader, and bad style, it is not incorrect.
Best Answer
Those are both correct. However...
Those are also correct1. What isn't correct:
You can't use just alone in that position after the subject. That has to be just now. However:
Those are fine, but have a different meaning, at least in the dialects I am most familiar with - just in that case has the meaning of barely. The other cases of just without now can also have that meaning, though it would be differentiated from the very recently sense by stress or context. Navigating dialects never being straightforward, I am informed that this use of just can also mean the same as just now in at least one regional dialect of British English.
1: Which is more natural or conventional is heavily dependent on dialect, and I don't just mean American English vs British English - regional dialects play a big part.