Well you're only gonna notice change in certain areas by doing pull ups :P And again depends on diet as well as varying in speed on different people.
Also varies per person how many you need to do, but see if you can do like 8-12 proper pull ups in a set. Then repeat another 3 or so times with breaks :P. Really I can't tell how many you are able to do, I have no idea what your strength is like lol. Remember technique is very important, and the better your technique you have the quicker you'll see change.
First off, make sure there is a strong frame that you are putting it on as I've seen numerous damaged walls/frames from them.
Second, what is the goal of this? General strength improvement? size gains? wellness? weight loss?
1.) Depends on your fitness level, preferably until failure. To see change you need to overload the muscles and while pull-ups use large muscle groups, you'd definitely see better results (as well as not being strangely proportioned) including this with other exercises. I'm unable to determine if by extra exercise this is your only exercise or if you have a routine and this is an addition. If the goal is to gain size, you will want to include other back and shoulder exercises, and if you are going to failure you won't want to be doing this as a daily routine (gains require rests to rebuild). So again the goal needs to be stated.
2.) You can realize some change in strength within the first 2 weeks if you do not normally use these muscles, however it will all be neural adaptations. If you are aiming to have a ripped back, depends on the shape of your body as it is now. If you have large stores of fat, time will be increased. And then factor in your body type (do you gain muscle easy or hardly at all) and your diet...
So overall, hard to say given the limited amount of information here.
general strength improvement and weight loss are the main goals, I already do a bunch of other exercises but recently have been looking into exercising a bit more when I have nothing to do or when i'm waiting for queues in games i'm playing, do a few push ups, lift some small dumbbells anything I can do in my room really. My diet and other exercise is fine ive already lost 20kg with it but want to build some muscle for when I get closer to my goal weight and was hoping getting a pull up bar would help increase weight loss and muscle gain at the same time which I heard building muscle helps burn fat faster?
First just after a week or two you will notice difference in strength, you will gain muscle as long as you get stronger, it can take months before you see any change, but get strong and muscle will follow.
if it's a weight issue then pull ups aren't the best to do, 20 a day will lose a couple pounds a week but also gain muscle at the end of the day
do pull ups and walk/run/bike ride for 20 mins a day for weight loss and muscle gains
EDIT: I notice you just want to do this in your room, keep doing what you are doing, add some squats and crunches - Jog on the spot (high strides)
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Also make sure you know the difference with pull ups and chin ups..
Pull ups are palms facing away from you with hands at shoulder with apart or more (but not too much as you'll just injure yourself as it gets harder and more straining for your joints. Pull ups work the most of your back chest and arms
Chin ups are aimed mostly at your arms, biceps.. this is the one with your palms facing your chest and chest width apart (these are far easier than pull ups)
Well if you want muscle keep the rep range between 7-12 in sets of 3 or 4 if you can handle it. focus on the muscle contraction, not just pulling yourself up quick and coming down quick, and controlling yourself.
You would see more significant and quicker results if you had a routine. 30-60 mins 4-5 days a week at home could get you real far in terms of weight loss and muscle gain. There are tons of exercises you could do with no equipment but a good investment I found was a resistance band.
Like most said, you'll notice strength gains within your fist few weeks. Depending on your body composition, the typical time to see your actual physical progress is a couple months, so don't get discouraged if you don't notice any change to your body the first month or so. And like Minrolol said, focus on using your back muscles. It's almost counterproductive if you just swing/kip your way up, as you're not getting the full benefit of using proper form.
And, like any other muscle, you're going to want to let those back muscles and lats rest for a day or so to repair itself. Rest is key for gains.
I applaud anyone who has the drive to keep themselves in shape.
Get an iron gym, no need to screw anything in and they are pretty strong, if they can hold my weight (17 stone) then they must be bloody good. Not to mention you can also use it for situps and pushups with adjustments, it takes 2 minutes to assemble then you are good to go.