Well, the CPU and GPU do share "main memory" so yes.
By creating a "config.txt" file on the SD-Card you can control (amongst many other things) the screen resolution, and thus the number of pixels the CPU has to shift.
You can also choose how much RAM goes to the GPU, by renaming one of the "GPU binary blob files" (firmware for the GPU) to "boot.elf", there are normally three one for each memory split, with 32MB, 64MB or even 128MB going to the GPU, and only one is simply named "boot.elf", that is the one that is used. Each of them supports the maximum resolution, so that is not a reason to not chose the 32MB one, but playing video, or using 3D graphics is influenced by the amount of available GPU RAM.
By the way, a much cheaper solution than using a HDTV with an HDMI interface is using a ("VGA") monitor that happens to have a DVI input as well, as is the case with many monitors that are used as VGA monitors. Even fairly old ones came with a DVI-D interface connector, and you only need a cheap MDMI to DVI converter cable to use them. The PI will also "negotiate" with the monitor for the default resolution, so you might even see a speed increase when using an old monitor. Just look at the back of any old VGA monitor you have, if you see (next to the VGA input) another slightly larger (often white) one with one larger rectangular input slot you are in luck. Google for DVD-D pictures to see how the connector looks like. DVI-I connectors are also okay, but not the (quite rare) DVI-A connectors (which are missing a lot of pin holes in the middle of the connector, but see the google pics to be sure)