I have created a small compost pit and i have putting lot of garden cuttings and kitchen waste in it for last 12 weeks. The whole thing has turned into nice dark brown humus. It smells very nice and I am planning to spread it in my garden.
Though I noticed that some of the twigs and peanut shells have not completely decomposed. Am I just being impatient in using the humus or should I wait for few more weeks till every single twig is decomposed.
This is being done in Pune (India).
It sounds as if the bulk of your compost is ready to use, so no don't wait, they could take a lot longer than you think. The best thing to do is riddle the compost so that the twigs and shells are removed, or simply pick them out as you spread it if its not a huge amount, and you can add those to your next pile. Twigs do take a long time to break down - it's much quicker if you can cut or shred them before adding to the pile in the first place.
No, there's no reason to wait to use the compost that is ready to use now. Different inputs to the compost pile take different amounts of time to break down.
We've been actively composting pretty much everything that can be composted here on our farm for the past 10 years or so and about the only thing that I'll do to twigs is cut them down a bit to make them more manageable in the compost pile (easier to turn with a fork) and to increase the surface area of the material. I don't have a chipper-shredder but I might be inclined to use one if I did.
For inputs which are slow to decompose, I'll just leave them in the compost bin and, over time, they'll break down. Our bins are dynamic things that are always "cooking away" at something.
To simplify this process, I use a coarse screen (1/2 inch hardware cloth or maybe some chicken wire) to screen out the larger chunks. I don't worry particularly too much about uniformity in size of the compost, other than to screen out the pieces that need more time.
You can increase the decomposition rate by decreasing the size of the carbon-rich materials like twigs by chipping them up or breaking them up. Again, it's about surface area. Chipped/shredded twigs will break down faster, all things otherwise being equal.
Since you've said that much of the compost has broken down nicely, it is clear that the decomposition process is working well. Again, the carbon-heavy materials take time sometimes, particularly dense materials. I've got one bin that has pruned twigs from last winter in it and they are partially broken down. I just sift those out and throw them back into the bin.
Aside: It wouldn't be terribly awful if the twigs/shells weren't completely broken down and they ended up in the garden. They'll break down there too. But I tend to leave the materials that aren't quite done yet in the piles.
Actually, it is beneficial to the soil to have some amount of undecomposed organic matter, because as it decomposes, it makes humic acid, which breaks up the soil particles into minerals the plants can use. I often add compost that is not completely done on purpose, for this reason.
But sticks are high in carbon, and so rob the soil of nitrogen when added in significant amounts. In your case, I think you'll be fine. I often use a different 'slow' pile for composting woody materials, because they are very valuable, but often take a lot longer to decompose than the other things commonly added to compost piles.
You can also bury your woody items under the "active growth" level of the garden and let them break down in situ there. Unchipped, they break down slowly, so they don't have as much of a nitrogen-robbing effect (and at a level below normal tillage, where you put them to have them out of the way when working the soil, that's less immediately relevant anyway.) Some folks I know have tried this in filling raised beds, and it appears to be working, at least in the sense that the crops don't seem to be nitrogen-starved "due to rotting wood" even though the lower half of the bed is pretty well full of wood, with dirt/compost on top.
