1) Lag can be caused by hardware, corrupted software, a rogue app, an app hogging too much CPU, malware and a few other things. IOW, anything inn the entire computer that's not working right.
2) Can malware have downloaded if you stopped the download? Yes, if the payload was in the beginning of the file and that part got through before you stopped it - and the downloading app didn't delete the partial download. Which is why having a good antimalware app and a good anti-virus app running when you're downloading is the best thing to do. Avast, for instance, traps downloads, checks them and, if they're okay, releases them as if it wasn't there. (You see the notification in the notification bar as it's checking.)
It's also a good reason to have everything backed up at all times. If it comes to it, you reflash the stock ROM, then restore your entire data partition, minus the file with the virus.