Ultrasounds in the third tri are highly inaccurate when it comes to sizing and dating. And by that I mean several weeks as a margin of error, which would be about right. Were you unsure of your dates? Did you get an early u/s in the first tri for dating? Those dates are highly accurate- within days. So unless you're totally unsure of your date, I wouldn't worry about that part or about the size of your baby (unless other things are going on like gestational diabetes or something).
As for contractions starting and stopping- that is very common. Your body is gearing up! Contractions are rarely useless, especially if they have been crampy, stretchy feeling, or painful at all. It sounds like your cervix is definitely making progress and that means when you finally do go into actual labor everything will be quicker (and hopefully easier). Measuring dilation is a bit of an art so different practitioners can measure differently. You can also dilate and undilate based on your physical and emotional state. So it's entirely possible that you were several cm dilated and then went back down. It is also entirely possible to get to 5-7cm without actually being in active labor (this happened to me with my second... I'm not sure how likely it would be with a first baby). Have you had other signs of impending labor like losing the mucus plug? Bloody show?
I know how exhausting it can be to have contractions start and stop, start and stop and to think "this is it!" only to be disappointed again. I also know what its like to be in a lot of pain toward the end of your pregnancy and to just want to get on with it already! But I have to just say, it would never be my choice to induce at the hospital if I could avoid it. Induction just so often leads one intervention after another and for first time moms especially, greatly increases your chance of a c-section. If I were in your shoes, I would look into more natural forms of labor induction first like homeopathics, herbs (like the cohoshes), nipple stim, walking, scrubbing floors on all fours, external castor oil, and as a last resort ingesting castor oil. But again, a word of caution, if your body simple is not ready to go into labor, attempting to push it along can often result in more frustrating starting and stopping. A good way to know if things are for real is to drink a full glass of water, lay down or take bath and if relaxing makes them come on stronger and closer together then that a good sign. If relaxing calms them down, then they are likely more warm up contractions.
Good luck mama!