Core 2 Duo's are obsolete, so we can rule that out. If you follow my posts on this forum, you'll find that I always support Intel. However in this scenario for a HTPC, perhaps you could go down AMD's route? AMD have been working on their APUs, which are integrated graphics built into the processor. Yes Intel have integrated graphics with their Intel HD, however AMD are superior in terms of integrated graphics power. AMD's A10-7850K with Radeon R7 (on-die) is capable of playing games such as Grid 2 at medium graphical setting at 50 FPS (
source). I understand you're not intending to play games, however that was to illustrate the graphical power of AMD's APU.
It's completely your call whether you want Intel or AMD; you should be fine with either as you're not planing to play any games. Even an Intel Pentium with Haswell architecture should be able to meet your requirements; if you're going with Intel select the currently supported socket, which is socket 1150. Any processor on socket 1150 should be adequate, even the Celeron/Pentium as mentioned earlier; as they have integrated Intel HD graphics and are dual-core.
If you're building a HTPC, you're going to want a small case and motherboard, correct? If so, and you want the utmost high-end AMD APU with an ITX motherboard with WiFi and dual HDMI you could go for the following (remember this is the utmost high-end AMD APU, which is reflected by the cost; you don't have to spend that much if it's out your budget and to be honest, I don't expect you to spend as much as below):
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD A10-7850K 3.7GHz Quad-Core Processor ($166.99 @ NCIX US)
Motherboard: Gigabyte GA-F2A88XN-WIFI Mini ITX FM2+ Motherboard ($92.99 @ Amazon)
Total: $259.98
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-08-16 20:21 EDT-0400
You can go with any AMD APU of your choice, but again Intel is always available. Any Intel CPU on socket 1150 should meet your requirements, even the Celeron would run circles around the Core 2 Duo you initially suggested. The Intel Celeron's on socket 1150 can be had for $45 (lowest price).
All the prices above are in United States Dollars (USD), if you reside elsewhere, please state your country and budget.
All the best.