Best smartphones in the budget of Rs.10000

Finding a phone that meets all your needs in a single device would not be an easy task. Let me list out a couple that almost meets your needs.

Yu Yureka Plus
It comes with a 5.5 inch screen. The processor is capable of a speed of 1.5 GHz. The phone carries 2 GB RAM and 16 GB internal memory. Yu Yureka Plus runs on Cyanogen OS 12 that is based on latest Android OS. Being a Cyanogen device, you can be assured of regular updates to the newer Android versions. Front camera on the device reads 5 MP and the rear camera reads 13 MP. The battery capacity of the phone is 2500 mAh.
You can buy it at Rs. 8999.

Coolpad Note 3
Coming from the latest entrant in the Indian smartphone market, it comes up with 5.5 inch screen at HD screen. Powered by a processor of 1.3 GHz clock speed, it has 3 GB RAM. It runs on Android 5.1 Lollipop. The rear camera has 13 MP resolution, while the front camera is of 5 MP resolution. The battery capacity of the smartphone reads 3000 mAh. Internal storage of Coolpad Note 3 is 16 GB.
You can buy the device at Rs. 8999.

As for using the Internet from PC on your mobile, I do not think it would be possible. As far as my information goes, you would need your phone to be rooted. There are some reverse tethering apps available on Play Store. Most of them need your phone to be rooted.
For using your PC Internet on your mobile, you need to employ a technology named reverse tethering. Phone in itself does not support such feature. You will need to install a mobile app for reverse tethering. Some apps need you to install a corresponding app on your PC as well.

The Meizu M2 or Metal if you can stretch or fine a good deal. Lenovo K3 Note is seems really good. Just see if your networks work so you can get at least 3G.

These are the smartphones which will satisfy your requirements.

1. Acer Liquid Z630s

With specifications of Dual Sim, 4G, 3G, Wi-Fi and with Octa Core, 1.3 GHz Processor powered by 3 GB RAM and with internal memory of 32 GB this is a powerhouse and with a monstrous battery of 4000 mAH and a wide screen of 5.5 inches with a 720 x 1280 px display and 8 MP Camera with flash and has Memory Card Supported, up to 64 GB which will run with Android, v5.1 ( Lollipop). This is truly a beast and you can go with it and the price is Rs 9,999, which is available in Flipkart.

2. Lava Pixel V1 (Android One 2.0)

With specifications of Dual Sim, 3G, Wi-Fi with a Quad Core, 1.3 GHz Processor powered by a 2 GB RAM, 32 GB inbuilt memory accompanied with a 2650 mAH Battery with a 5.5 inches screen, 720 x 1280 px display. With 13, MP Camera with flash and has Memory Card Supported, up to 32 GB which will come with an Android, v5.1.1. This is available in Flipkart for Rs 8,499 and in amazon and paytm for Rs 8,999.

3. XOLO Black 1X

With specifications of Dual Sim, 4G, 3G, Wi-Fi powered by Octa Core, 1.3 GHz Processor with a 3 GB RAM and 32 GB inbuilt memory accompanied by a 2400 mAH Battery. With a 5 inches, 1080 x 1920 px display and with 13 MP Camera with flash and also has Memory Card Supported, up to 128 GB which runs with Android, v5.1. This Smartphone is available in Flip kart for Rs 9,499 and in amazon for Rs 9,750 and in Snap deal for Rs 8,999.

The following phones that are priced under Rs.10000 have earned good user rating till this March 2016 which are based on Android, IoS and Windows Operating System and have the best combination of features, with good performance and value for money:

1. Coolpad Note 3
Coolpad Note 3 scored a 9/10 rating for its value for money. The Android 5.1 smartphone has a 5.50-inch Display with a Resolution of 720x1280 pixels, a 1.3GHz Processor, a 5-megapixel Front Camera and Rear Camera of 13-megapixel. The phone comes with a RAM of 3GB and a Storage of 16GB with a Battery capacity of 3000mAh. The features like fingerprint sensor, battery life, camera are quite satisfactory. But the UI customizations are a little disappointing.
Price:Rs. 8,999

2. Yu Yureka Plus
The Yu Yureka Plus secured an overall rating of 9/10 for all its sections. Running on Cyanogen OS 12.1 Processor clocked at 1.5GHz, a Display of 5.50-inch and a Resolution of 1080x1920 pixels, it has a 13 megapixel rear camera and 5-megapixel Front Camera with 2GB RAM and 16GB Storage with a 2500mAh battery.
Price: Rs. 8,999,

3. Xiaomi Redmi 2 Prime
The Redmi 2 Prime from Xiaomi got a respectable 7/10 rating because of its camera and design. Coming with a Display of 4.70-inch with a Resolution of 720x1280 pixels, the Android 4.4 OS phone runs on a 1.2GHz Processor. The Front Camera is of 2-megapixel with a Rear Camera of 8-megapixel. The phone has a RAM of 2GB with a Storage of 16GB with a Battery capacity of 2200mAh.
Price: Rs. 6,999

4. Lenovo A7000 Turbo
Lenovo A7000 Turbo is a Dual Sim phone with 13.0 Megapixels Camera with Flash. It is an Android 5.0 - Lollipop phone that comes with a 5.5 Inch Display, and a resolution of 1920 x 1080 pixels. The rear camera is of 13.0 Megapixel and the Front camera is of 5.0 Megapixel. It has a powerful 2900 mAh, Li-ion
Battery that gives a Talk Time of 36 Hours.
Price: Rs.9,295

5. Xolo Black 1X
This Dual Sim phone accepts NaNo-SIM and GSM type if SIMS. The OS is Android 5.1 - Lollipop that runs on 1.3 GHz , MediaTek MT6753 Octa Core Processor with a RAM of 3 GB. For GPU it has chosen ARM Mali-T720 MP3, 450 MHz. It comes with a Rear Camera of 13.0 Megapixels Camera with Flash. The phone has a 5.0 Inch Display with 1920 x 1080 pixels of Resolution.
Price: Rs.8,999