1998 Hyundai Accent Car shakes when idling

Hi I have a 98 accent GS 3-dr and when I am in neutral or when I am driving and I clutch and brake to come to a stop, the engine starts to shudder, about every 2 seconds. The rpm goes down to what sounds like 500 or lower (almost stalls but hasn't yet) then jumps back up to normal. It doesn't always do this and temperature doesn't seem to affect it. My clutch seems to make the car less smooth to shift (don't know if its related to the engine studdering) I've driven other cars of the same year/model and slowly letting off the clutch in 1st smoothly moves the car whereas mine its all or nothing and you got to give it gas as just the right moment not to rabbit hop it through the stop sign.

One other question, I had some almost new winter tires from an old car and I used them on this one. Car takes 155-80-13 and these are 175-70-13 or something like that. Is it normal for my car to sway back and forth? (Not as slow speeds, I mean on the highway when steering moves are smalll (keeping the car in lane/lane changes)

Thank you so much

-Shawn