I usually play in Eb standard. My strings have the treble strings from a pack of 9's and the bass strings from a pack of 10's. I am thinking of tuning to D standard (for a long period of time, not just for one song), and was wondering if there's anything I need to do/not do if I'm going to do this. Will the strings be too slack?
no, you'll be fine. although u might want to get thicker strings for d standard
you should be fine.
you 'll just need to adjust your intonation and maybe your truss rod if need be.

PS. you can get hybrid sets of strings.
call me crazy, but i recommend 12's for 1 step down. they'll feel like they're in standard but the notes will be 1 step down. if you have a floyd rose bridge you'll have to do a lot of adjustments for any of which you've mentioned but if you don't you should be fine.
definitely wouldnt be ideal but if the strings dont feel and sound too limp to you than youre fine. its really just a matter of personal preference. if you want the guitar to be in that tuning all the time you should re-setup the guitar for that tuning, especially the tremolo if you have one. and with strings that slack it might not be possible to get the intonation spot on at all (not that any guitar is ever perfect) with the lower tension. i would recommend trying a bigger guage (ernie ball makes a hybrid set one bigger than the slinky top heavy bottoms youve been using that would be like trebble strings from a set of tens and bass strings from a set of elevens) if you dont want to change the feel.
dude youl be fine, ive done that before. As long as you dont keep tuning from e to d then back and forth, etc.
string gauge to me is dependant on the scale length of the neck.

i use 12-56 on a 25.5" scale tuned to C standard.