Would this confession be valid? If a priest prematurely says the prayer for absolution, is the confession still valid?

Is the confession still valid even when the priest is not wearing his normal vestments? Or if I didn't say the prayers the priest asked me to pray until I got home because I couldn't focus during the mass?

Is it necessary for a penitent to bring up unconfenssed sins the next time they receive the sacrament of reconciliation?
A few things concerning your confessor from the Catechism:

The confessor is not the master of God’s forgiveness, but its servant. The minister of this sacrament should unite himself to the intention and charity of Christ. He should have a proven knowledge of Christian behavior, experience of human affairs, respect and sensitivity toward the one who has fallen; he must love the truth, be faithful to the Magisterium of the Church, and lead the penitent with patience toward healing and full maturity. He must pray and do penance for his penitent, entrusting him to the Lord’s mercy.
CCC 1466

The sacraments comes from Christ, not from the person you are going to confession with.  Vestments, etc.. don't make it valid - it might make it illicit; then again, I don't think the things you mentioned are as grave as some of these things that make the sacrament truly illicit.
It is certainly OK to postpone the satisfaction that the priest has given to you as penance. I've never confessed much of anything that didn't involve going and saying a few prayers.  I have confessed things where I had to pray an entire Rosary (being quite a prolific sinner).  But the priest wouldn't expect me to do for that before Mass.

The penance the confessor imposes must take into account the penitent’s personal situation and must seek his spiritual good. It must correspond as far as possible with the gravity and nature of the sins committed. It can consist of prayer, an offering, works of mercy, service of neighbor, voluntary self-denial, sacrifices, and above all the patient acceptance of the cross we must bear. Such penances help configure us to Christ, who alone expiated our sins once for all. They allow us to become co-heirs with the risen Christ, “provided we suffer with him.”
CCC 1460

And the priest can give you some penance that requires more than a few seconds between Confession and Mass to perform; you're still absolved under the normal criteria.
That's what is important - if you are sorry and you confess your sins, they're all forgiven.
However (and this comes from the Handbook of Prayer's guide to confession that I've read a few times), if you remember some sins you didn't confession in previous confessions it is good to confess them at your next confession.  And it is even OK to bring up past sins that you have confessed, as context for the priest.
For instance, if you had a problem with pornography, but you watched an R-rated movie with nudity (as opposed to an X-rated one) then you shouldn't feel prohibited from mentioning that - it helps the priest to know what and why you're confession what you're confessing.
And, I'm not going to give pastoral advice (because that's verbotten on this site) but priests who don't show they're taking your confession seriously are probably telling you that they aren't the kinds of priests you want to visit to develop a habit of frequent confession.