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.