If I go with a construct, will I have trouble with the wardens?

Not likely. As long as it doesn't look/act/smell/etc. like a zombie, or any other walking violation of one of the Laws, and you don't go around using it to kill people you should probably be fine.

Will it be alive?

Not technically. It can seem alive, but it won't be alive in the sense of possessing a mortal soul.

Emotions and the whole gambit?

That entirely depends on how complex a construct it is. A low-end model with only a modicum, or less, of free will can't even think for itself, while a high-end model sporting the magic equivalent of advanced artificial intelligence might be capable of learning, thinking for itself, philosophy, emotions and the whole nine yards.

how will the construct function in combat situations?

This depends on what you make it of, and what you make it for. Something built for combat will likely function very well in combat, while something built to be utilitarian might be abysmal in a fight. As for controlling it, short of thinks-for-itself AI, it should simply follow your orders.

Will it work on my character sheet like an enchantment?  Maybe several enchantments?

That is entirely up to you and your GM to sort out what it counts as, if you wanted it to function as something like an ectoplasmic, or even native materials, construct which you're only able to summon to you once in awhile then this might be the way to go. Or maybe even an enchanted item a la Bob's skull which serves as the home of a spirit bound to your will which you can send out to animate/possess objects and the like.

Most importantly, how do I find out how much refresh my construct will have?  how do I find out if my character is even capable of making one in the first place?

More than anything else, this one is entirely between you and your GM to sort out. Just keep in mind that the better the construct, the more difficult it would be to make.

"On A Steel Horse I Ride"
Invoke: Your favorite transportation is your trusty motorcycle; invoke as appropriate for anything involving said bike.
Compel: You absolutely insist on using your bike even when another vehicle might prove better suited to getting you someplace; also the usual loss/damage of item stuff

"Wanted: Dead or Alive"
Invoke: You're a wanted man; invoke for bonuses to social skills, particularly intimidation, and knowledge of various criminal things
Compel: Someone, or something, wants you real bad; compelled for trouble with cops, being hunted by your pursuer, etc.

"Karma Chameleon"
Invoke: For luck and karmic bonuses.
Compel: It comes and goes; sometimes your luck just runs dry and leaves you in a tough spot.

As it's written, anyone (mortal) with even a single casting power, like Ritual (Ectomancy), is subject to the troubles of accidental hexing. The flipside is that those people can also engage in deliberate hexing.

The Overflow rules are on YS 214, but they expressly bar you from taking offensive actions (which applying Aspects to other people, i.e. Aiming, is). Further, the rules for using a spell to apply Aspects are different from the rules for doing it with a Skill, namely that by Spell the extra shifts you kick in to duration don't make it sticky, they just make it last for that many more exchanges at which point it simply poofs (obvious exceptions, like lighting something On Fire, not withstanding). As written, evocation maneuvers say "...a temporary aspect...," it's only when you get into Thaumaturgy that it says you can slap on more than one at a time.

Aww... I saw the title and wanted to say something about a vacuum cleaner  or a bucket of water (beings of "smokeless flame" and all...)

But in all seriousness, it depends on which take of Djinn you want to go with. Djinn as another name/kind of angel might make things twitchy and difficult, but in typical Dresdenverse it would seem that they're completely unrelated so that part's up to you and the player.

As for playing as a Djinn, just stack up some Nevernever powers (Swift Transition and Gaseous Form come to mind), maybe a Recovery/Toughness with a Catch of holy stuff, and possibly even Channeling or something.

For a master of a bound djinn, I'd err on the side of treating like an enchanted item, but more mechanics than that I'd need some time to work out.

REDCAP

The Catch (+3) is cold iron and such

Given their mythical penchant for wearing iron-shod boots and fighting with iron weapons, that Catch might be rather debatable, but that's just my take.


Siren
High Concept: Siren Temptress
Other Aspects: A Voice Sweeter Than Honey; To Listen Is Madness

Skills:
Great: Athletics, Deceit, Performance
Good: Alertness, Presence, Rapport
Fair: Conviction, Discipline, Endurance
Other Social skills, and Stealth, default to Average, rest to Mediocre

Powers:
Incite Emotion (At Range of Sound, Lasting Emotion) [-5] Lust/Attraction
Human Guise [-0]
Wings [-1]
Claws [-1]
Feeding Dependency (flesh) [+1]
-Inhuman Speed [-2]
-Incite Emotion (Potent Emotion) [-1]

Stress:
Physical OOO
Mental OOO
Social OOOO
Hunger OOO

Total Refresh Cost: -9

Notes: Greatly disdain combat and if forced into a fight will take to the air and assault their attacker's mind with song.

Oh, hmm.  That hadn't occurred to me, but I see what you mean.  I guess I'll just get rid of Echoes and just go with Beast Change by itself.

Actually... I might have found a way for you to keep Echoes, and the Human Form bonus, and pick up Claws for the fox form.

Item of Power (Hoshi no tama) +1
-Glamours -2

Human Form +1
-Inhuman Speed -2
-Claws -1

Then leave the rest of your powers and stunts as they are.