This trip would require the same travelling arrangements/expenses as doing the trek to base camp though, right?
Yes, but you are looking for something cheaper than $10 a day (which is what a typical trek there cost)? Few places in the world are cheaper than this. You could cut down more by skipping any extras, but it's kind of pointless unless you are really destitute.
If so, is there perhaps a more distant (from Everest) and closer (to perhaps an airport or main town) area to climb that would cut down on the travelling costs?
You don't have to go all the way to Kala Pattar to see Everest as you can see it long before that.
Just out of curiosity, how long would the trek to Kala Pattar take, approximately? And how much would the overall adventure cost?
Average cost is about $15 per day (less than $10 without a porter) including all lodging, food and porter or porter/guide (guides are not needed, but it's nice to get a porter). You can get by without a porter, but to me it's worth the cost and helps out the locals. Porters/guides typically charge $5 a day or in some regions up to $8. It's better (or at least more flexible, cheaper and safer) to get a porter/guide rather than to go through an agency, at least if you aren't doing any climbing. You can do it without a porter guide, but it's a small price to pay for someone to carry your pack and it helps out the local population as well as provides a more cultural experience.
Total cost for the trek itself (food, lodging, *porter) should be $225 for 15 days + round trip airfare Kathmandu to Lukla (about $225) or if you walk in from Jiri total cost for the trek itself should be $315 for 21 days + one way airfare Lukla to Kathmandu ($120) unless you have the time to walk all the way out.
$550-$600 plus airfare to Kathmandu should be more than enough including the trek, hotels and in Kathmandu, shopping, permit, extra luxuries, etc.
An average budget would be $10 a day and you can live pretty good on that. Spending more than $15 is pretty hard.
*If you have a choice, a porter is preferable over a guide. The popular trails in places like the Everest region are very easy to follow and navigate. If you can navigate a typical hiking trail in the US, then you can navigate to Everest Base Camp. Some books have maps describing the entire route in lots of detail. The one below is really good:
http://www.amazon.com/Trekking-Everest- ... 891&sr=8-1
Your profile doesn't say where you live. California? Airfare out of LAX to Kathmandu for example, runs around $1000 round trip.