Anyone with some guidance? Attached my project as an image below...
So, I have done few testing and got very interesting results. And want to do more. And I came across with the idea of bringing Arduino project out of the development board and run it standalone using Atmega328P U micro-controller chip.
OK, I think you have got matters a bit confused here.
The Arduino UNO as illustrated above is not a practical form. To some extent it is indeed a "development board" or perhaps more appropriately a "demonstration" or "experimental" form. But the alternative is not to make a "standalone" version.
For serious "real world" projects, what you want to use is either an Arduino "Nano"
or if you do not need the USB interface and can program it using an adapter, a "Pro Mini"
These are ready-assembled and ("clones") inexpensive so you can connect to them either by soldering to the terminals, or soldering the pin headers to them and mounting them either to stripboard or your own custom-made PCB. There are also "screw shields" and other shields to spill out the various connections using "Dupont" cables.
What makes no sense is to try and purchase all the necessary parts separately. Even though the regulator on these boards is essentially useless and you may not want the "pilot" LEDs (or USB interface chip on the Nano) if operating from batteries, it is generally cheaper to purchase the ready-built modules which are a reasonably well-designed and tested PCB pattern using SMD parts.
How can I handle the power with 9 of total LED's (Normal LED's for now) including Ultrasonic Sensor.
Well, you use a regulated 5 V power supply, don't you? (Do not try and use the "Barrel jack" on the UNO or "Vin"/ "Raw".)
I have to place the sensor approx. 6 1/2 meters away. I'm hoping to use a telephone cable between the sensor to the circuit. Do I have to plan a circuit to manage the current with some components?
It's not the current, it is the need to supply sufficient signal over such a distance. In respect of an ultrasonic sensor, it makes more sense to locate the Arduino reasonably near the sensor and run less critical connections - to LEDs - over the cable. "Cat 5" - readily available - has 8 wires, so power and 6 LEDs. It might be better to use two separate Pro Minis and serial communication between them, but if you do not need all 9 LEDs to be on at once, then six wires is probably sufficient.