No, you don't always need to use a period or semicolon. However, in order to look at your sentence, I need to go into some detail about it—because it also touches on another point of grammar.
Here is simple example:
He stumbled and he fell.
Also, the two things that are joined together don't necessarily have to be related by any kind of cause and effect:
He ate pretzels and it rained.
This is not something that would normally be expressed in this way, but it's still understandable.
In the first sentence, it's arguable if there are two unrelated things going or if it's a single compound action.
In the second sentence, there are two independent clauses joined by and—but they are succinct enough that there may be no reason to use a comma to separate them.
The Chicago Manual of Style (17th ed.) says this:
When independent clauses are joined by and, but, or, so, yet, or any other coordinating conjunction, a comma usually precedes the conjunction. If the clauses are very short and closely connected, the comma may be omitted (as in the last two examples) unless the clauses are part of a series.
And here are the two examples it refers to as being correct:
Electra played the guitar and Tambora sang.
Raise your right hand and repeat after me.
Note that in my previous example, eating pretzels and it raining aren't related to each other at all; nonetheless, it might still be considered acceptable to omit the comma because of the extreme brevity of the sentence.
Here is your own example sentence that's been shortened (for simplicity) and with consequently removed:
He did not submit the application by the deadline and his application was not considered valid.
The use of the coordinating conjunction, without punctuation, serves the same grammatical function as in the example sentences above.
Adding consequently back into the sentence, there would typically be two normal ways of punctuating the sentence.
1. Consequently is restrictive information, and a comma is used after deadline for the sake of clarity (because the second clause is actually not very short):
He did not submit the application by the deadline, and consequently his application was not considered valid after it got dispatched to the admissions office.
Note that an alternate phrasing could also be used:
He did not submit the application by the deadline, and his application was consequently not considered valid after it got dispatched to the admissions office.
2. Consequently is considered nonrestrictive information, and is therefore separated by a pair of commas:
He did not submit the application by the deadline and, consequently, his application was not considered valid after it got dispatched to the admissions office.
So, your sentence with consequently is just fine—except that you would normally want to put a comma after deadline. Also, the comma that you use after consequently may or may not need to be there—depending on if you want to use it restrictively or non restrictively.