Bud, im offering helpful advice here, only you can see your engine mounts at this particular time. Lol. Removing the insert will resuce vibration yes, but with it in it will not be vibrating bad enough to cause your wheel and door to rattle (unless doors etc are loosely bolted on?)
The way you described it, it sounds to me like a broken engine mount! The inserts go front and back, and if only he front is in it will not cause anything that isn't annoying. With the rear it will be firmer but not really as you have described. Ive Seen so many members buying inserts to put in and then selling them on, as they are finding they have them fitted without even realising! Haha thats how un imposing they are.
The way yours sounds, and the way you described where it was coming from is what would happen if your drivers side engine mount has failed...or if there is a torque damper that has been fitted and it had vibrated loose....these are two things you will need to check first by looking. Literally lift the bonnet, look at the left of the engine, if you see a split engine mount it's buggered...needs replacement. (There will be fluid as described above)
If that looks good, check for the torque damper if one is fitted ...a blue and gold piston located at the back left of the engine bay (next to the brake fluid). These connect the engine to the car and are renowned for vibrations that are annoying.
If it doesn't have anything blue and gold there, and the engine mount on the drivers side is not split then you could try removing the insert at the front (from underneath) and try. You will soon find that there will not be a massive difference. The integra is a fairly firm car so vibration should be expected. If it seems abnormal still then check what i have advised above and see. Speaking to the seller may help, but actually looking will soon identify if there is an issue or not bud. There isn't much else i can suggest.
Failing all of that, take it to a garage and see what can see/ feel.
If its not doing it when moving then my money is on engine mount drivers side.
I don't mean to sound abrupt, but just see if you can see anything. Fire up a picture of you like and ill try and helps from that...without being there to see it, feel it etc.
Good luck.