The fun way to do this, as well as the obvious interpretation of the rules, would be to have them imitate phrases perfectly, sounding like the original speaker, but using them in approximately the sense they originally heard them. So their voices can change completely from sentence to sentence - one being a dwarf they heard speaking yesterday, the next sentence being a small elf child they heard ten years ago, and the third being something you said to them three months ago.
They understand what phrases and sentences mean, but if they can't break down phrases into words and use the words individually, they're still fairly limited by their curse. They would sound a bit like Dilbert's pointy-haired boss, who tends to come out with phrases and clichés that belong in a subtly different conversation.
This would be an interesting challenge for a player who was usually verbally adroit. Precedents for even more extreme versions exist in the Librarian from Terry Pratchett's Discworld series who manages to get by with just one word, "Oook", and Mr Punch in the classic Punch & Judy show, most of whose lines are "That's the way to do it!"
The main problem with playing a Kenku this way would be avoiding humour in serious moments of the game.
No, not always.
When mimicking the speech or sounds made by another person, yes, you gain advantage on your Charisma (Deception) check. You could benefit when you mimic the speech of an Dwarf, or the war cry of an Orc.
When mimicking the sounds made by by "other creatures," no, you do not gain advantage on your Charisma (Deception) check. You could not benefit when you mimic a horse, bird, or the roar of a hydra.
Ultimately it's going to come down to what your table decides a "person" is in the context of the feat. It may be "any creature of type humanoid" (borrowing the implicit definition from hold person, though no rule says it should or should not apply) or it might be "any intelligent creature" such as an Ogre, Fiend, Celestial, Dragon, or other creature with human-like intelligence.
My interpretation would be that it applies only when mimicking humanoids or the speech-like sounds of intelligent creatures. So, you could benefit if you mimic the serpentine lisp of a blue dragon while speaking, but not the earth shattering roar of that same dragon no matter how hard you tried. (Not that you couldn't attempt it, merely that you would get no advantage due to the feat.)
Still another interpretation is that you'd be required to be trying to pass yourself as a specific individual to benefit from advantage. So mimicking the noise an Orc makes when he bellows wouldn't benefit, but mimicking the Orc guard from three weeks ago who bellowed when you stuck him with your sword would gain a benefit. That's an awful narrow reading, however.
There's many potential interpretations here, and none of them are necessarily more or less "right" than anything else.
Ultimately it doesn't really matter as long as you're consistent.
Best Answer
Anyone can attempt to do that
Anyone can attempt to do anything, whether they have the feat to do it or not (even try to recall everything accurately, as in Keen Mind). It doesn't mean they will always succeed, but it also doesn't mean they need the feat to make an attempt.
This is an example of bad wording, for a feat
The way this feat is framed implies that only characters with this feat can succeed. Drawing on the wisdom from the Unearthed Arcana: Feats, specifically on the "bad feat" (as identified in the article) named "Warhammer Master," we learn the following:
This is in response to this part of the Warhammer Master feat:
The design goal, as far as Mike Mearls was concerned, was to write feats in such a way that they did not limit the game's flexibility. The way this part of the Actor feat is written implies you need to have the Actor feat to even attempt to mimic someone's speech.
How do non Actors do it without stepping on the feat?
I believe the non-obvious but valid intent, which will still grant this ability to everyone else without necessarily needing to step on this feat's toes, was to imply that the Actor is so good, they can mimic anyone after one minute's worth of study. Everyone else has to take at least a few hours before they can even try properly.
Listen to a chirping bird for a minute, or a bubbling brook, and try to mimic that sound. You might be able to, but you probably can't; but, someone in the world probably could (out of the 7B people on Earth, chances are pretty high, right?)
This feat makes your character so good, they will find that person is themselves, for any sound or speech pattern.
What do they roll?
The Actor feat gives us the guidance for what skill to use:
This makes sense as you are trying to deceive someone that you are what you are not. Based on this feat, then in most cases, you should roll Deception vs Insight. However, for non-Actors, you could use different skills as you see fit.
A Charisma (Performance) check could replace Deception, if you interpret this particular check as a test of how masterfully they can mimic another sound. Perhaps they are openly mimicking robins singing to an audience of children.
A Charisma (Stealth) check could replace Deception, if they are trying to hide by creating noise that makes them blend into the environment.
If interpreted this way, then for the Actors, the predictability of what skill to use means they know where to put their proficiency and Expertise, should they acquire them. This makes this feat even more useful, because for Actors, they always know which skill to use.