Learn Italian: Lui copia sempre un'idea qualsiasi. - He always copies any random idea.

While qualsiasi usually goes before the noun to simply mean "any" (as in qualsiasi idea - "any idea"), placing it after the noun changes its nuance. After the noun, it translates more closely to "random", "ordinary", or "mediocre". So un'idea qualsiasi emphasizes a lack of originality: just some random, run-of-the-mill idea.

If he copied multiple random ideas, would qualsiasi change its ending?

No, qualsiasi is completely invariable, just like the adjective altrui you are focusing on in this lesson. However, there is a strict rule about its placement with plurals: when placed before a noun, that noun must always be singular. It is only when placed after the noun (meaning "random" or "ordinary") that it can be used with plurals, as in delle idee qualsiasi.

Since Italian drops subject pronouns, why is Lui included at the start of this sentence?

While not strictly necessary, including the subject pronoun Lui adds deliberate emphasis. The speaker is likely contrasting him with someone else or putting a spotlight on his specific behavior, highlighting that he is the one who always resorts to copying.

Is the position of sempre fixed, or could it go elsewhere in the sentence?

Its placement right after the conjugated verb is the most natural, standard position for adverbs of frequency in Italian. You could technically put it at the very end of the sentence to put a heavy, frustrated stress on the word "always", but the current word order is the most common and neutral.