Taking it off or not will be your choice. I personally never take my fitbit off unless it is to charge it. I still use a HRM as well for certain workouts. I find the calories to be pretty accurate for runs and walking so I don't use my HRM for those anymore (generally get pretty close to the same number calorie burn for the time period from both devices).

When it comes time to log your workout, you will be asked for the start time, duration, and calories burned. As long as you put the correct start time and duration the calories you log will overwrite whatever the fitbit had tracked for that time period.

I don't log any exercise calories on MFP. I find it causes too much confusion with the fitbit adjustments. I just let them sync. If I have to do an adjustment, Ill do in by entering an exercise on the fitbit app.

As for your heart rate monitor, I am betting the biggest difference you will see is things like weightlifting, where you are mostly standing still, and biking, where the fitbit doesn't pick up the motion.

I use the Fitbit app to log my workouts for the same reason. However, I suggest you don't use your HRM for weightlifting since it's not steady state cardio. It's kind of like using a kitchen scale to measure lengths.

The exercise data should go both ways, I log it on Fitbit just because there are many more options (i.e you can pick a different effort or resistance level for the same exercise). I logged it on MFP once and I was double credited, I'm not sure why (perhaps I did something wrong).
I'm not familiar with Rushfit, I guess it's cardio-based interval training? An HRM might not be ideal but it's probably your best bet. I would trust it for a few weeks and see how you're doing: you can always readjust your calories if you're not losing at the predicted rate.

I thought you were supposed to log exercise on MFP and not fitbit, since the exercise data syncs from MFP to Fitbit but not the other way around!

I'm not really doing straight up weightlifting, I'm actually doing the Rushfit program (similar to P90X and Insanity but more oriented toward fighting/martial arts and functional fitness). It involves free weights too (up to 20lbs) and many exercises that don't involve moving your legs at all. And as part of the program, I also jog 1-2 times a week.

So I'm guessing the HRM will be more accurate than the Zip in logging the Rushfit workouts, and probably the jogging too (though in the jogging it will get much closer).

I log my exercise to Fitbit if it is an activity I need to log. You can do it either way. MFP suggests logging here because then the specific activity appears on both sites. If you log to fitbit, the extra calories you burn (beyond what MFP expected) is transferred over form Fitbit but not the specific workout. I have a few reasons that I prefer logging to Fitbit directly (I think Fitbit's activity database calorie burn numbers seem more reasonable, I don't like being too dependant on the timely synching of Fitbit and MFP to compile my total calorie burn, it simplifies the process of following the "accounting" on MFP). Either works, it is just preference. Sometimes the sync between the two is pretty instant and sometimes there are delays. I like to have all my calorie burn in one place and all my calories eaten in one place--in case of any delays as I can work with that more easily than working with incomplete information.

About weight lifting or really any type of strength training. Yes you want to log it (probably). The problem with using a heart rate monitor to estimate the calorie burn is that hrm's are not designed to estimate calorie burn for non-aerobic activity. They are said to be less accurate for cardio intervals too. Resistance training isn't aerobic exercise though sometimes circuit training and other methods of training may have some aerobic benefits. It is more like interval cardio though. Calorie burn isn't strictly based on heart rate. In a lab, they use oxygen consumption. HRM companies use formulas to estimate your oxygen consumption based on what percent of your maximum heart rate you average in a workout. They estimate calorie burn from that and from your stats. I always read that HRM's tend to give inflated estimates for anaerobic exercise (including weights) and for intervals (usually using anaerobic work). I am not convinced that is strictly true, since most HRM's are using your workout "average heart rate" vs. listed "maximum", I think it may depend how fast or slow your heart recovers. I tend to see very low calorie burn estimates for intervals on my HRM. I've recently taken to logging non-aerobic exercise using the Fitbit database. I use to log it with my HRM and that worked okay for me with weight control, but really I don't think it was that accurate.

I would probably log the more vigorous RushFit workouts as "calisthenics, vigorous effort" or "circuit training" (which will actually give the same calorie burn, I use the calisthenics if it is mostly bodyweight i.e. pushups, burpies, etc. and circuit training if it is mainly weights just to keep it straight in my mind later). The flexibility workout, I would probably log as yoga. anyway, logging it this way does increase my calorie burn more than logging my heart rate monitor.

I still think it is a little more accurate since I would go to a very vigorous circuit training class where we did circuits of heavy weights and vigorous calisthenics with cardio intervals as our rests--burpess, mountain climbers, jumping rope, etc. By the end of the hour I would be very red like I had a sunburn, all my muscles exhausted, dripping with sweat. When my HRM app logged it to fitbit, this activity would count as "moderately active" (based on my HRM's calorie burn estimate). I would get more very active minutes from walking to and home from the gym than from this class. This was a very vigorous workout--the most vigorous I did at the time. So I tried logging it as circuit training on fitbit, and it credited it with very active minutes and added about 70 more calories or so to my allowance. It was actually this class that really made me realize how imperfect a heart rate monitor can be for resistance training. I did okay eating accordingly when logging my HRM burn because it was usually a very low, conservative estimate. But for someone that it overestimates for, it may not work as well. If you want to continue logging your HRM burn, fine (I did for a couple years so can't judge), just be aware it is an off-label use and keep an eye on how it effects progress towards your goals.