Buy every bit of bathroom fittings you need . Working ‘ backwards ‘ from fitting too ‘ see ‘ how the pipework should be is much easier .
Solvent weld is the stuff . If you aren’t brave that it sets in 3 seconds use tiger seal ( or similar ) gives lots more play time .
Personally with waste from basin/shower/bath - unless it’s an exceptionally long run then any fall would be suitable . Await @Nickfromwales to tell me off .
if your 110mm isn’t far away ( like mine , cause I actually planned that 😁 ) it’s unlikely to be an issue .
Now stop fannying and posting and do it !