"Siempre me siento cansado."

yes, some verbs need the indirect object as a pronoun before it.

I agree that this should also be accepted. The gerund form of the verb here implies that it's something unpleasant that is happening more often than one would desire.

because it is "I always feel tired"

"Always I feel tired" is not the way most English speakers would say it. -unless maybe the person was very upset about it, and said "ALWAYS I feel tired!" -even then, most English speakers would say "I ALWAYS feel tired!"

If someone asked you "Do you feel tired today?" you might reply "I feel tired always." Otherwise its usually "I always feel tired."

They wanted the "I" first. The way you said applies to poetic or Old English grammar. Not the kind of order you say casually.

It is, but it doesn't go with modern grammar. It's something you would hear from the 19th century, and definitely not something you say a lot.

I am sorry I don't recall the sentence it was in.