Couple Problems

Problem One:

Whenever my car is idling, whether hot or cold, and i rev the engine once it drops back to idle like it should...but then a couple seconds later it dips real low and sounds like it's going to stall. Is this normal?

Problem Two"

When I try to put it into reverse it is very rough, sometimes it even grinds the gears. I shift into it just like any other gear but it just seems to be difficult.

Does anyone know how I can fix these two problems? And even if i have any? Thanks a lot!

Yes i bought the car 100% bone stock, and it doesn't only happen when it is cold...it also happens when it is at normal operating temps.

Oh and i found that putting it into reverse was easier if i put it into first before, but i havent made it a habit. I will start doing that though!

Thanks!

EDIT: Oh and I bought it with a broken heater core, I replaced it in a casual 2-day time period along with the help of my Dad so im hoping that as long as we have the right tools that we will be able to fix any problem that arises.