secular bands playing modern hymns
He does not like the band that plays at times during some liturgies. Why he can't stop them, I don't know.
(Note: Playing music in a Catholic band myself, this part of my answer might not be unbiased.)
The first sentence actually contains two completely different statements:
The first statement is that you don't like the type of music played in the masses in that parish. However, you should keep in mind that you probably won't find any comprehensible argumentation why a certain kind of music is more suitable for being used in the mass than other kinds of music. In the Bible, we find a lot of completely different musical instruments that were used to praise God.
The second statement in your first sentence is that these bands are "secular". This would mean that these bands are playing songs whose text (!) is not related to praising God. I doubt that this is the case.
I know that a lot of very old songs that have been used in Catholic masses for centuries (here in Germany) use melodies that were taken from secular songs. So if you call some music "secular" if the melody is taken from a secular song, "secular" music was used for centuries in Tridentine masses!
Your second sentence is related to the first statement:
What would a "Tridentine priest" do, if he does not like traditional music and thinks that "modern hymns" (as you call it) are more suitable for the mass?
Would he "stop" using the traditional songs in the Tridentine masses in his parish?
If no: Why do you think the priest in your parish would act differently?
A special form of praying I also know from some parishes.
If I understand some statements in answers to this question correctly, faithful touching one another while praying were already mentioned in the Bible and the form of praying you describe is traditionally used in other Christian denominations.
As far as I know, the Catholic mass (both the Tridentine and the new form) is full of forms of praying that have been invented by polytheistic religions and taken over by the Catholic church. (This is one reason why Jehovas Whitnesses refuse the Catholic church!)
So if you don't mind using forms of praying invented by polytheistic religions in a Tridentine mass, why do you mind using a form of praying invented by another Christian denomination in a mass?
At least this form of praying seems not to be officially forbidden - even not in a Tridentine mass!
the faithful in immodest dress
and I see people casually approaching the altar.
I doubt that such people were a rarity in Tridentine masses before the Vatican II Council.
My personal impression is that such people prefer the "mainstream" and therefore do not visit (I can't use the word "to attend" here) Tridentine masses but Novus Ordo masses.
If Tridentine masses were the mainstream, you would complain about such people attending Tridentine masses.
There is no kneeling for the Eucharist
An older priest told me years ago, that this even differed from country to country before the Vatican II Council.
If this is true, there were also Tridentine masses where no kneeling was done for the Eucharist!
However, if you think kneeling down is more respectful than standing:
I have often seen people kneeling down during the Eucharist prayer and also for receiving the sacrament of Eucharist in parishes where all other people are standing. I never saw that that that was not respected by the other faithful.
... make a mass invalid?
Thanks, I am struggling with this. Please help!
Once again your question has two completely different aspects:
The first aspect is if there is some dogma or similar written by some theologian saying if such a mass is "valid" or "invalid".
The second aspect of your question is if such a mass is "valid" in the eyes of God.
I think that the second aspect is the one that you are actually interested in.
However, there is nobody who can answer you this question.