Exactly the same. "אני נועלת נעליים" can mean that you have your shoes on or that you are putting them on.
They are the same word. The extra י is put the compensate for missing nikkud.
Except that "shod" is an adjective. It looks like the third form of a verb, but you never use any other form. You might say "I am shod", but you'd never say "I am shoeing"
It's quite correct about the English "to shoe ..." as a verb (past tense would be shod). It's not usually used these days for humans, but as you pointed out, shoeing still applies to horses:-)
Hebrew verbs are usually three-letter roots. In this case:
Other examples: שמר, כתב (keep/guard, and write)
The infinitive form gets a lamed prefix and a vav between the second and third root letters (in the simplest binyan):
לנעול לשמור לכתוב
The present tense forms for male/female and singular/plural have a vav between the first and second root letter and a suffix that matches the gender and number:
I don't get it, it says both of those words mean wearing shoes. Must those two words be always used in conjunction? Can you use one or the other?
The second word in the sentence is a verb. It is translated as "wearing", but it's more specific in that it only refers to shoes. The same verb has another meaning, "locking", but the point is that you can't use this verb for wearing a shirt, a hat, trousers, socks, and the jury's still out on boots.
The third word is the sentence is a noun -- shoes. So "wearing shoes" translates as "נועלת נעליים". You need both words because they are different even though they are derived from the same root.
Suppose you omitted the noun and only said, "אני נועלת", you might be understood, because only shoes (and doors, and locks, and safes) go with this verb, but that would still be an awkward way to say this.
Interesting- is there an alternative verb for wearing boots, then? ("Suited and booted" is a set expression in English to mean well-dressed and well-prepared) Oh, and can one use "נועל" for sandals?
According to the Hebrew academy (link below), the verb נעל is to be used for shoes, boots, sandals, clogs. There are no special verbs for different footwear.
Surprisingly, there is an informal verb from the word sandal: סִנְדֵל (sindel), but it is not used with any kind of footwear, and its meaning is "to hinder" or "to thwart". Its origin is from the 1980s when the Israeli police used Denver Boots for a short while, and in Israel they were called סנדלי דנוור - Denver Sandals. They have been out of use for 30 years, but the term remains for generally blocking or thwarting someone.