Short answer, "No" You might have some residual discomfort after the "novocaine" wears off. It might be due to the actual injection and/or the drilling of the tooth, but if there is any discomfort, it should resolve very rapidly. If not, go back to your dentist. Sometimes the bite needs a little refinement, or there are other underlying problems that have to be addressed, especially in deep, extensive fillings.
You will feel sore. You may feel a little bit of soreness and sensitivity to hot and cold temperatures immediately after the novocain (procaine) wears off and maybe for a day or two. Any discomfort is very temporary and will quickly wear off. Take Ibuprofen or any otc pain medication in the meantime.
No. Your dentist will most likely use a topical anesthetic to numb the gingival tissues prior to injection.
Thanks for asking! You would fill a pinch with the shot as well as some pressure/burning in the immediate period when the local anesthetic is injected. After that, it will be much better and you won't feel much of anything.