Everything you want to do can be done with any laptop - except gaming. That is always a problem because you will need a high performance GPU which produces a lot of heat. And the heat problem is one of the main drawbacks in laptops because there is little room for adequate blowers and to evacuate the hot air.
Rather than focussing on the CPU, I would focus on the GPU in your case. An i5 CPU is a good compromise and plenty for most applications. It can even handle such tasks as video encoding pretty well. For the GPU you have to look what is being offered, but it should have at least 1GB of it's own RAM. The so called on-board video facilities will not do for gaming.
I have equipped all my systems (including my 4 laptops) with SSDs. I would never go back to spinning disks. But the best is to buy the SSD seperately and exchange it against the HDD. That recovered HDD you can use as an additional external storage with the help of a USB enclosure. They cost pennies.
Speaking of USB - you should have at least 1 USB3 port. That makes external disks a lot faster. An eSata port is even better. I once had a Sony laptop that had both. But that cost $1175 and may be far out of your budget range.
I would recommend a 120GB SSD and put some of the data on an external drive (e.g. the enclocure or large USB sticks). But if you plan for very large games, a 256GB model may be better. You are looking at $100 to $200 for those. Apart from the SSD performance aspect, you also get a lot less heat from an SSD than from a HDD.
Bottom line: buying a laptop is easy if it were not for the gaming. A desktop has a lot more options and can be equipped with a lot more powerful devices than a laptop which is limited an space and therefore very heat sensitive.