So you know how to program drum machines and program synths but know not of how "remixes" are made? or what a sampler is?
Technically a sampler is so you can make your own snips of audio like the ones you use in the drum machine. You don't 'need' one as you can do it all in the DAW you are using. Some people like to use one because of workflow, creativity or the unique sound the piece of hardware will apply to it. A rompler synth is a synth filled with samples, Nexus for example. Some drum machines are samplers too and some samplers can be used as a drum machine.
As for vocals, yes, do it by ear, this is music, sound lol. Some people build a track around the vocal after adjusting its bpm, pitch or just leaving it, others make a track and look for a vocal to fit, then make more adjustments.
There will be a time stretch function in your daw. Figure out the bpm of the vocal using this formula, create a loop, count the beats in the loop (imagine them), Divide the number of seconds in the loop by the number of beats. Divide 60 by the answer. You now have a rough BPM for that loop, the longer the loop the more accurate the answer. Or, use a beat calculator where you tap the imaginary beat as the vocal plays. Put the bpm of the vocal in your time stretch tool and the bpm you want, Press GO! Job done. Sometimes you want to preserve pitch other times you don't but that's enough for today.