1st things 1st:- Both Tachnun & Hallel come from the same book of Tehillim; so Mimonofshoch, both on Yom Ha’Atzma’ut & 23 days later on Yom Yerusholayim, we are all going to recite the same book of Tehillim, some of us from the earlier Perek 6, and other from latter Perkim 113-118.
By a Shiva, we also are all going to recite the same book of Tehillim, some of us from Chapter 16 {or 49 according to some Minhogim} whereas other will still be reciting Chapter 6. By a Shiva we don’t recite Tachanun because Tachnun reminds us of Din, and by a Shiva in the immediate aftermath of the departure of a beloved, who cannot be thinking of the ongoing Din of the recently departed individual!! I hope this answered your 2nd question, Gubbish.
As for your 1st question Gubbish, your very 1st question should have been:- How can I Daven at a shiva, since there is a Halocho:- That 1 should always Daven at his/her Mokom Kavu’a, and not be Minyan jumping around. I was taught many years ago at a Halocho shiur, that despite this Halocho, sometimes other considerations override this Halocho, such as the Chessed of being “Menachem Avail”. Clearly, not having a Minjan at a Shiva house, and even having to await a Minjan, is a Bizoyon both to the Deceased’s Neshomo & to the Availim, whereas a large crowd is a huge Kovod both to the Niftor & the Availim, so you Gubbish should absolutey attend the Shiva Minjan, eventhough you will be omitting Tachnun. Reciting Perek 16 {49} and a Mishna Shiur will probably take more time than Tachnun, anyways.
By attending the Minjan at the Shiva, not only are you not being bad about Tachanun, but you are being a Tzaddik for fulfilling and complying with a Torah obligation of “VeHolachto BiDerochov” :- “Mah Hu Menachem Avail >>> Gam Atoh Menachem Avail!!!”