You can definitely submit an offer without a buyer's agent (we just did this successfully with no issues) - the listing agent should have you sign paperwork that they are not representing you, but they are just facilitating the transaction.
REALTORS® are negotiation experts. If you buy a home without an agent, you'll have to negotiate and decide how much to offer on your own. This may cause you to unknowingly overpay for your home – or lose out on one you want.
If the buyer is not working with a real estate agent and does not have representation, the offer would come directly from the buyer. Represented buyers can also write their own offer letter and have their agent submit it for them.
Yes, you can write your own contract. However, including all necessary elements is crucial to make it legally binding.