Hi Kosh,
1. Budget: As mentioned by Andrea, you would need a Sagarmatha entry permit, 30$ and TIMS card from KTM. TIMS would cost you 20$ (for FIT, Green Card, assuming you are trekking individually). I never had any problems trekking independently in that region, not sure if rules have changed since few months back.
Apart from these two permits, you should account for around 10-15$ for cab from KTM airport, and 160$ for the air tickets to Lukla.
You would have to add expenses for buying and renting gear, food, rent for accommodation (if you are not very particular on accommodation quality, many places would waive off your staying charges if you have your food there).
2. Equipment: From your trail head to EBC, the weather would change and you would have to carry low-altitude/warm as well as high-altitude/cold clothing. Around October, it should not be as cold as in peak winters, so you would not need heavy winter clothes. Your favorite trekking shirt+pants should be good till Namche, IMO.
From there on, you might eventually need thermals under your clothes, jacket, headgear (something that covers your ears, cold winds during that time can be a pain) and gloves. Some rain protection, just in case. Attire can be brought at Thamel according to how you feel once in KTM.
For the actual trek, you would definitively need a (preferably two) shock absorbing hiking poles, 35-40L backpack+cheapo rain cover (possibility of last rains in October), trekking boots (don’t get anything that does not have ankle support) + the usual trekking stuff like map, water, etc.
3. Toughness: You might have heard that EBC trek is probably one of the easiest BC treks, but that is when you are talking about trekking in the Himalayas. Overall, the trek can be quite strenuous. You would be walking 120 odd kilometers over 15 days, ascending 4000m from KTM, and getting back down, temperatures can be around 32 in KTM afternoons to -2 in BC. That being said, on a tight budget & trekking independently, you do not need a guide or a porter for this trek. I am assuming you would be fine with carrying your stuff which would be around 25-30KGs.
Many times you can find a porter from en route places, but don’t count on it. The trail is quite clear. Proper mapbook and cellphone GPS (with offline maps) works perfect for me.
My training is jog+walk 15KM every morning with 20KG backpack on slight uphill. If you are comfortable doing something similar to this at least 3 months before your trek, you are definitely in good shape. As for climate/temperature/altitude, no real way of preparing for it. You have to gradually adapt to it on your trek. No concerns if you do not rush your trek. Take as much time as possible, there are tons of things to do and see on this trek.
Best of luck!