Learn Icelandic: Ég borga alltaf fyrir áskriftina. - I always pay for the subscription.

No, you cannot. In Icelandic main clauses, the verb must stay in the second position (this is called the V2 rule). Therefore, the adverb of frequency alltaf (always) has to come directly after the conjugated verb borga, making it Ég borga alltaf.

What case does fyrir trigger in this sentence?

Here, fyrir takes the accusative case. When fyrir means "for" in the sense of paying for or exchanging something, it always forces the following noun into the accusative. That is why the object is in the accusative form áskriftina.

What is the base form of áskriftina and how is the word "the" added here?

The base noun is áskrift (subscription), which is a feminine noun. Because fyrir puts it in the accusative case, we need the accusative feminine definite article attached to the end. For feminine nouns ending in a consonant, that ending is -ina, giving us áskriftina (the subscription).

Why does borga look exactly like its dictionary infinitive form?

Because borga (to pay) is a regular verb belonging to a very common group. For most verbs whose infinitive ends in -a, the first person singular (the ég form) in the present tense is completely identical to the infinitive. So, "to pay" is að borga, and "I pay" is ég borga.