I am very late to this party but hear is my $.02
Removing the magazine disconnect can be as easy as removing the retaining pin in the trigger without removing the trigger and shaking the gun until it falls out of the magwell or it can be require a hydraulic punch. If you are at all hesitant about the skills required to do this vs your gunsmithing skills I highly recommend getting it done by a good smith who is familiar with the BHP. It will cost you a lot more if you mar your BHP frame.
With FN/Brownings there are no hard and fast rules or dates as to which one you own. In general on Pre MKII guns C, T and earlier guns you can remove the small pin in the trigger which retains the magazine disconnect and the parts will drop out. Once you get to the MKII you have to remove the trigger assembly. This is where it gets a little tricky. MKII guns are generally easier than MKIIIs. Bllued guns are generally easier than epoxy finished guns. The worst are late model MKIII with the Epoxy finish. Some of these require a lot of force to get them out. I have seen guns where "gunsmiths" even marred the frame when attempting to remove the pin.
As others have stated the pin is directional. Right to left. If you do it the other way you can enlarge the hole in the frame which will result in a loose trigger pin which will walk out. Use the correct tools for the job. If you don't you might regret it. Make sure you tape off the area of your frame that might come into contact with the pin. If you bugger your frame there are solutions for this if it does happen but why be that guy. If it does happen to you get this pin.
Hi-Power "Self-Tightening" Trigger Pin Solution | BHSpringSolutions LLC
I highly recommend this tool from BHSS. It is like $20 shipped to you and will help you out. You will also get a free recoil spring with your purchase if you have the code.
Hi-Power Trigger Pin Removal Solution | BHSpringSolutions LLC
If you have a MKIII I suggest trying to remove the pin but if you run into a lot of resistance soak trigger pin area of the the frame with ballistol and let it penetrate the area. Then try it again working carefully. The BHSS videos on this are one of the better ones out there.
Now as to how much it is going to improve you trigger pull there is no definitive answer. In my experience just the removal of the part reduces the trigger .5 lb to 1.5lbs depending on the gun. Lost of people tell you it dropped X pounds but don't have scale and are going by feel. Don't believe them. What they are feeling is less drag and resistance and a smoother trigger which they translate to less pounds. Mags will now drop free and that is the only really reliable result of the mag disconnect removal.
Now for the downside of the mag disconnect removal. It can effect the reset of the pistol. It will become even more vague than it is in stock configuration which pretty vague vs say a Glock 19 or 1911. I recommend replacing the trigger return spring while you are in there to a heavier spring which will give you a more positive reset. Since you are in there replace the hammer spring with a 26# which will have more effect on the trigger weight than the mag disconnect. Some people go lighter but if you shoot the gun enough you will get light strikes and eventually many develop hammer follow if other specs are not adjusted to run that low a spring. All these springs can be bought from BHSS.
*This might sound like an ad for BHSS but it isn't. I am not a huge fan of them for anything but springs and tools but that is a topic for another day.
As to the origins of the mag disconnect it was part of the original contact request made by the French when the approached FN to build them a new pistol. They wanted it so that a gun should be presented for inspection and made safe by simply removing the mag. They also like the idea of a more safe mode because the French Army was conscripted and only marginally trained which was true of most Armies at the time.
They never adopted the pistol but the Belgians did in 1935 and they kept the magazine disconnect. The only guns ever to leave FN Herstal without them were some of the guns made under Nazi occupation of Herstal and they removed it for speed of production and because they did not deem it necessary.
Also remember how much you paid for my $.02 so take it with a grain of salt but some people say I know a bit about BHPs