Is a lintel required once bricks removed for a under stairs doorway

If there is a load above (More brickwork or stair load) then yes you would do. You'll also need to apply for Building Regs for the formation of a WC.

Yes you would if it was a load bearing wall above the doorway you intend to create. This would support the bricks above the doorway & the stairs.

In most cases yes a lintel would need to be used. However the size/material required would be down to a structural engineers evaluation. In most cases a new WC will require building control as mentioned in previous answers. The structural engineer can submit his calculations as well to building control for confirmation and the builder then can follow and have signed off the structural works before boxing/finishing the area. The general things the structural engineer will be looking for are: number of floors above, load bearing walls, vertical loads, size of door way, type and construction of property and/but not limited to the points the lintel will bear onto (ie will it potentially need pad stones if the opening is big enough to require a larger lintel/steel). Local building controls are always the best first point of contact to advise what they require and what they think. Normally a small consultation is free be that via email/images or a small quick site meet.