Bonzo 
Sooo, gums. My least favourite bit of dentistry but arguably the most important... you wouldn't put a roof on a burning house so they say, meaning unless the gums are healthy, it doesn't matter what you do to the tooth.
I'm not sure where to start so bear with me.
The teeth sit in a special type of bone called "alveolar bone", the sole purpose of this bone is to hold your teeth in. If there were no teeth, we would just have the jaw bone and this is why, if a tooth is removed, the bone will resorb as there is no longer any need for it. They are held in place by the periodontal ligament in a special sort of joint called a "gomphosis". On top of this you have the gum which sits around the tooth and seals off the tooth (and the mouth) from the bone.
The gum around the tooth creates a sort of cuff, usually about 1mm deep. This is called the gingival crevice, it constantly leaks fluid which is why placing fillings under the gum can be hard, it's impossible to stick stuff there as it's always a bit damp.
Over the course of the day or night, plaque bacteria (that live in the mouth constantly) sit in the gingival crevice and carry out their routine business of secreting bacterial toxins, multiplying and creating something called a biofilm. The effect of all of this on your gums is to cause "micro ulcers" in the gingival crevice. The gums become red and sore and they bleed when you brush (or eat etc) because the ulcers are being rubbed. This is gingivitis and it can form pretty quickly, within 24-48 hours.
The best remedy for gingivitis is a really good clean, with a toothbrush. This will remove the plaque from the gingival crevice and allow the gums to heal. Our default reaction when something is sore and bleeding though is to steer clear of it, this is completely the wrong thing to do with gingivitis, get amongst it with a really good clean and it will almost certainly resolve within 12-24 hours.
The reason mouthwash isn't especially effective is twofold. Firstly, imagine the gingival crevice as the line around a shoe, where the shoe joins the sole. If you walk around a muddy field and want to clean said who, just sloshing a bucket of water over it will probably make a little difference but it's not going to pass close inspection. The mud will have washed off the smooth bits but it's probably still stuck in the awkward bit between the shoe and the sole. You need to get n there with a brush and manually scrub away at any left over muddy bits.
The second reason is the biofilm , essentially this is where bacteria have one over on us. Biofilms are everywhere, notably in pipes and tubes. We use chemicals to try to get rid of them but a biofilm is an impenetrable mass of bacteria, protein and slime. Chemicals (like mouthwash) are entirely useless. The only thing that can remove a biofilm in the gingival crevice is manual cleaning. With a toothbrush.
If you brush effectively twice a day, your gums shouldn't bleed. If they do then you shouldn't just go and buy corsodyl, you should see the dentist who will hopefully send you to a hygienist because they are amazing at spotting what you're not doing and remedying it. They may be pricey but they are cheaper than new teeth!
More to come on gums, I'm just going to have my supper.