The costing and timelines of a UX project can be given typically in many methods. Someone with a project management background will give in more detailed like resource level billing, adding overheads, travel and other costs. This all will be put in the contract of work.
Startup design company might cost a project in the different way as they want to keep the client happy, work on something interesting and deliver the best output.
General rules of costing are to see that you get paid for the efforts you put in and never underpaid. No need to see competitors but think of a figure which gives you comfort. Depending on the scope of the work you will have to play smart to win the project and then plan.