Learn French: Tu regardes la télévision tout en faisant le repassage. - You watch television while doing the ironing.

Both indicate that two actions are happening at the same time. However, adding tout emphasizes the simultaneity of the actions, making it clear that you are doing the ironing at the exact same time as watching TV. In other contexts, tout en can also add a sense of contrast or concession (like 'even while'), though here it purely stresses 'at the same time'.

Could I use tout en faisant if someone else was doing the ironing while I watched TV?

No. The gerundive (en + present participle) always shares the same subject as the main verb. Since tu is the subject watching TV, tu must also be the one ironing. If someone else were ironing, you would have to use a conjunction with a new clause, such as pendant qu'il fait le repassage.

Why use faisant le repassage instead of just the verb repassant?

Both are grammatically correct, so you could absolutely say tout en repassant. Using faire le repassage (to do the ironing) is just a very common idiomatic phrasing for household chores in French, similar to faire le ménage (to do the cleaning) or faire la vaisselle (to do the dishes).

Is there a liaison between tout and en?