Translation:My dog keeps getting sick.
"My dog is always sick." Would this be also acceptable? If not, then why not? Does 'always' always have to be "总是" ?
生病 is “to get sick/to develop a sickness”. This is why we translate present tense “to be sick” when the Chinese 生病了 uses perfect aspect (“got sick/developed a sickness” --> “is sick now”).
The difference between 总是 and 一直 is difficult to define… 一直 feels more like “all the time, continuously, perpetually”.
"Getting sick" sounds better to me, but I'm sure it depends on where one lives. Grammatically speaking, both are perfectly fine.
So apparently 一直在 and 一直 both mean the same thing.
The 在 is completely optional.
To me 在 does fulfill a function: It’s basically the difference between “My dogs always gets sick” and “My dog is always getting sick.”