On Yon Sheini and Yom Chamishi, I am usually not able to complete saying Tachanun with the Tzibur. Do I skip parts or continue saying Tachanun after Kriyas HaTorah – which would mean me not saying the remainder of the Tefila with the Tzibur.
You should say as much as you can before the tzibbur begins to say the seated part of tachanun (“vayomer david el gad”). When the tzibbur sits down to say the second part of tachanun (the part recited every day) you should skip to say this part of tachanun with them.
A possibility is to say one half of tachanun on Monday, and the other half on Thursday.
Having skipped part of tachanun, there is no need to continue after kriyas ha-torah.