Should people stand or kneel during the Eucharistic Prayer?
This is answered in the General Instruction of the Roman Missal, number 43:
In the Dioceses of the United States of America, they should kneel beginning after the singing or recitation of the Sanctus (Holy, Holy, Holy) until after the Amen of the Eucharistic Prayer, except when prevented on occasion by ill health, or for reasons of lack of space, of the large number of people present, or for another reasonable cause. However, those who do not kneel ought to make a profound bow when the Priest genuflects after the Consecration. The faithful kneel after the Agnus Dei (Lamb of God) unless the Diocesan Bishop determines otherwise.
Some people hold hands during the Lord's Prayer, while others hold their hands out like the priest. Is there a prescribed posture for the Our Father?
No position is prescribed in the Roman Missal for an assembly gesture during the Lord's Prayer.