Can you plant it outside, in the real ground? Or at least move the container outdoors? Mother Nature gives these plants so much more of the "right stuff" than we can do indoors. If you decide to move it outdoors, move your plant to a sheltered location outdoors to let it get used to the sun, wind, rain, and more before planting it in the ground or moving it to a more exposed location.

FataMorgana

Your soil looks too moist and the plant looks like it is suffering from damping-off. Also, I wouldn't fertilize unless, when the plants are more mature, they appear to need something, like maybe a shot of nitrogen if they're looking pale or yellowish. Many herbs prefer a lean, or poor soil, so you may be killing your catnip with kindness.

fatamorgana - I place my plant at the window, which i open during day. I cant place it outdoors as my cats will destroy it.

noinwi - is there any ways i can save it now? Should i give more coffee grounds since it provides nitrogen?

I turned a hanging basket wire frame over my baby catnip plant outside. With a couple of stakes . Cats had a good time & it grew well.

The plant in the photo looks like a goner...damping-off destroys the stem at the soil level. I would gently transplant any other seedlings that sprout into fresh potting mix without fertilizer and keep it barely moist, not wet. Get some chicken wire and fashion a dome-shaped cover or cage for it(similar to what Trythrice mentioned), so that when it gets larger, the cats can chew on the leaves that grow outside of the cage and the main part of the plant will not be harmed. This what I do for my cats during the winter inside. I place the plants outside during the growing season as they grow better in sun and develop stronger essential oils. Catnip really prefers full sun, so if you want to place it outdoors, you need to harden the plants off as Fatamorgana stated, by placing them out in the shade first, then gradually get them used to more sun until they are in full sun. Take at least a week to do this. Or, you could place a protected pot of soil outside and sow more seeds if you have time(you don't say where your gardening zone is).

noinwi - What type of potting mix/soil do you recommend for catnip? I live in Singapore, it is a tropical monsoon climate and it has been raining for the past few days so it is quite impossible to place it outdoors. I've placed the pots right next to the windows which they will get sunlight.

I think in your case, any type of bagged potting mix/soil made for houseplants(not one with added fertilizer)would do. Just remember not to keep it too moist and make sure the pot has good drainage.
Hope this helps, and let us know of your progress.

I had to make wire cages for my catnip when I first planted it to keep the cats and red foxes from eating it down to the roots. Once the foliage grew above the cage I removed it and the plant has reseeded itself enough for all so I do not use the cages now.

I think we have another case of killing with kindness here. Firstly, think of your plant as a baby. You wouldn't feed a baby on a rich adult diet and you don't give a seedling fertilizers until it is a bit more grown up. Potting mixes for seed sowing don't contain fertilisers for a reason so don't add any.
Secondly, think about where catnip grows naturally - it likes well drained soil in the sun. So it doesn't need constant watering. The mould and compost gnats both indicate that the soil is too wet. It needs fresh air and light. It won't mind rain as long as the soil it is in drains well.
I think you can save yours if you put it out in the ground. If you really can't then scrape off the fertiliser and coffee grounds. Stop watering for a bit. Give it a breezy spot in the sun and make sure it can drain. Protect it from your cats with some sort of cage. Good luck.

Just a small update, 2 of my catnip died after i put under the sun (behind a slightly tinted window).. 1 of it started curling upwards and the under leaf turns purple, and it is totally purple.

I bring them back in and shine with my lamp, and the curl was gone, but the under leaf is still very purple. Is this normal?

It can get very hot behind a window on a sunny day. On a window ledge in sun is not the same thing as being in "full sun" outdoors where there is cooling breezes, the coolness of the soil, and more to help moderate the temperature. Your little plants were probably baking on the window ledge.

Please go back and search through the back posts. Alot of people try growing herbs of one sort or another indoors with mediocre success at best. I wasn't just giving you a quick answer when I suggested moving the plants outdoors if you could. It really is the true recipe for success.

FataMorgana

fatamorgana - I have searched through posts and other website about growing catnip, but no one had the experience of growing it in tropical monsoon countries at the equator where the whole year its either sun or rain.

My worry is that 34 deg C in the day and the scorching sun comboed with torrential monsoon rains will kill the herb which is commonly grown in seasonal climates. In Singapore, even at cloudy days you can still get sun burn.

I am going to give the outdoors a try. I understand that there is a need to harden it, but what are the symptoms of pre-mature hardening and how do I know whether it is ready for the full sun?