Από το πρωί έχω βήχα και φτερνίζομαι συνέχεια.

It means since this morning / since the morning.
Literally, από means from, so Greek uses from the morning where English usually says since this morning.

This phrase gives the starting point in time for what follows: the coughing and sneezing began in the morning and are still happening.

Because Greek normally uses the definite article much more often than English does.

So where English might say since morning, Greek idiomatically says από το πρωί.
Here, πρωί is a neuter noun, so the article is το.

Also, after από, Greek uses the accusative case, and for this noun the accusative singular looks the same as the nominative: το πρωί.

Because Greek often uses the present tense for something that started in the past and is still continuing now.

So with a phrase like Από το πρωί, Greek says:

• έχω
• φτερνίζομαι

even though in English you would often use something like I’ve had or I’ve been sneezing.

This is a very common difference between Greek and English:

• Greek: present tense
• English: often present perfect or present perfect continuous

Greek can express symptoms with έχω + noun, just like English sometimes does:

• έχω βήχα = I have a cough
• έχω πυρετό = I have a fever
• έχω πονοκέφαλο = I have a headache

There is also a verb related to coughing, but έχω βήχα is a very natural way to describe the symptom itself.

So the sentence is focusing on having a cough as a condition, not just on the act of coughing once or twice.

βήχα is the accusative singular form. The dictionary form is ο βήχας.

So:

• ο βήχας = the cough
• έχω βήχα = I have a cough

It changes because βήχα is the direct object of έχω.

This is very normal in Greek:

• έχω φίλο
• έχω χρόνο
• έχω βήχα

That is a very common learner question. Φτερνίζομαι has middle/passive-style endings (-ομαι), but here it has an active meaning: I sneeze.

In Greek, some verbs are used in these endings even when they are not really passive in English. So you should learn φτερνίζομαι as the normal verb for to sneeze.

So:

• φτερνίζομαι = I sneeze / I am sneezing

It does not mean I am being sneezed or anything passive like that.

Here συνέχεια means continuously, constantly, or all the time.

It is functioning adverbially in this sentence, telling you how often the sneezing happens.

So φτερνίζομαι συνέχεια means:

• I keep sneezing
• I’m sneezing constantly
• I sneeze all the time

Be aware that συνέχεια can also be a noun in other contexts, meaning continuation or sequel, but here it is clearly the adverbial use.

Because Greek usually omits subject pronouns when they are not needed.

The verb endings already show the subject:

• έχω = I have
• φτερνίζομαι = I sneeze / I am sneezing

So εγώ would only be added for emphasis, contrast, or clarification.

For example:

• Εγώ έχω βήχα, όχι αυτός.
  I have a cough, not him.

In a normal statement, leaving out εγώ is the natural choice.

In this sentence, και simply means and, linking the two symptoms:

• έχω βήχα
• φτερνίζομαι συνέχεια

Greek και can sometimes mean also / too, depending on context, but here it is the basic coordinating and.

Greek word order is more flexible than English word order, because the verb endings and case forms give extra information.

So this sentence could be rearranged in some ways, for example to emphasize the time phrase or one symptom more than the other. But the original order is very natural and straightforward.

The given version works well because it starts with the time frame:

• Από το πρωί
  and then gives the two problems:
• έχω βήχα
• και φτερνίζομαι συνέχεια

So yes, word order can change, but this version is a very normal, idiomatic way to say it.