How often have two bowlers taken six wickets each in the same ODI or T20I?
And have two Tests ever played out simultaneously in the same country?

The Ireland-Zimbabwe and England-West Indies Tests briefly overlapped in the first instance of two Tests happening simultaneously in one country • Cricket Ireland
You're right that the one-off Test between Ireland and Zimbabwe in Belfast and England's third Test against West Indies at Edgbaston were being played at the same time. Unlike Dublin, where Ireland played their inaugural Test in 2018, Belfast is in Northern Ireland, which is part of the United Kingdom, so two Tests were thus being played in the same country at the same time. This has never happened before, although there was a near-miss in India in 2019: India played Bangladesh in Kolkata from November 22-24, then Afghanistan took on West Indies at Lucknow on November 27.
The unfortunate Clive Madande had a largely forgettable Test debut against Ireland in Belfast last month. First he bagged a golden duck when he batted on the first day, then conceded 42 byes in Ireland's modest total of 250.
You're right that all 11 of Ireland's players made a contribution in the field to their exciting win over Zimbabwe in Belfast last month. There's only one other Test in which all 11 players on one side took a catch, by New Zealand in Wellington in 2022-23. Actually New Zealand went one better, as 12 men - including substitute Will Young - held on to a catch over both Sri Lanka's innings. There are 19 instances of ten players taking a catch in a match, but no other 11s.
There's a grand total of one white-ball international that fits the bill here. In an ODI in Greater Noida in India in March 2017, Rashid Khan took 6 for 43 as Afghanistan (who had earlier made 338) restricted Ireland to 304. You might expect Rashid to do this sort of thing, but the identity of the Irishman who did well with the ball earlier in the day might surprise you: it was Paul Stirling, whose offbreaks brought him 6 for 55. He's taken 37 wickets in 160 other ODIs. There have been no such instances yet in T20Is, or in women's white-ball matches.
It's true that the Indian legspinner Riyan Parag's first three wickets in internationals - in his fourth T20 match, against Sri Lanka in Pallekele late last month - were all out bowled. But Parag has a fair way to go to break the record: the Pakistan seamer Khan Mohammad (in Tests in the 1950s) and Luxembourg's Ankush Nanda (in T20s against Turkey and the Czech Republic in 2019) both started by bowling their first seven victims. The England left-arm seamer Frank Foster (in the 1911-12 Ashes) and Bhutan's Tenzin Wangchuk (in T20s between 2022 and 2024) both began with six.
"Just to point out, you missed one in your list of two-country players who also appeared against their former countries: one of Boyd Rankin's Tests for Ireland was against England." You're right - Rankin won his only England cap in Sydney in 2013-14, then played in Ireland's inaugural match against Pakistan in 2018, and against England at Lord's the following year. That means ten of the 17 two-country players played Tests against their first team. Apologies to all for the slip - especially Boyd Rankin!
Steven Lynch is the editor of the updated edition of Wisden on the Ashes