Man, I scared you once... hopefully I don't scare you again.
Yes, especially as a subintern, you're expected to preround on your patients. You shouldn't wait to be asked. "Your patients" varies between rotations, as does what you're expected to do as part of your prerounding.
At my school, on surgery "my patients" was all patients on the service, but the expectation was only that I had numbers (vital signs, meaning ranges over the last 24h, trends if there was a lot of variation or an intervention, or in any unit patient), had done a very quick exam (laid eyes on, pressed on the belly, made sure limbs were warm sort of thing, + "passed any gas" "had a bowel movement" "any nausea, vomiting, constipation diarrhea, chest pain, shortness of breath?"). Took about 2 minutes per patient. I would write my notes as a I went or on the elevators, but usually had them done before the intern showed up (about five minutes before rounds). Would then scribe the plan as we rounded.
On medicine and neurology, "my patients" were the 3-10 patients I was following. I was expected to know a lot more about the patients, have done a very complete exam, and have a far more extensive note written before rounds, and come up with a plan. Takes about 10-20 minutes per patient (depending on how long I had more than anything else, but also if I didn't know what to do I would read up on it so I had a plan).
I wouldn't write "cheat sheets" if you can get away with it -- it takes time. I always presented from my note. One of my medicine attendings wanted all the notes in the chart before rounds, so I'd just photocopy them.
Some students think prerounding is useless scut, but prerounding is the best defense against DISCOVERY ROUNDS. That's when you're rounding and you find the patient "went missing" last night and hasn't reappeared, or wasn't preoped, or hasn't been beta blocked because the nurse didn't feel comfortable... or whatever. This tends to result in the intern looking bad (which is bad for you, because your job is to keep your intern looking like a star), the chief resident loosing their ****, and rounds draging on for a really really long time as the intern is chewed out for every single bad thing discovered on rounds... Not only is this boring after a while, but it adds at least 2 minutes per patient, usually more, while everyone figures out what is going on.
And, you get to take part in making the plan, because you're the first one to actually see the patients. Just make sure you know what to do at the hospital you're at if you find a patient in extremis, and your intern will appreciate knowing about anything abnormal before you present it on rounds i(so and so spiked a fever, yes cultures were drawn, no nothing is back yet; the guy in 62 is fine; pt in 83 didn't get his CT head last night but is scheduled for this a.m., no I didn't call but I'll do that now, yessir I know I should already have done it; &c .)
As far as how much time you will take per patient, try to get at least two-three patients assigned to you your first day assuming you're not on call, write some of your note the night before, and plan 45 minutes per patient since you'll be figuring out where labs are and such. Worst comes to worst, you'll be hanging out for a bit before rounds and you might be able to help your intern out with some scut. You can then adjust as necessary.
Best,
Anka