Flags:

The Dual Monarchy did not have an official national flag, instead you could see a number of unofficial flags being used.

In Cisleithania the most common was the traditional Habsburg black and gold horizontal bicolour often coupled with 'Landfarben', that is the traditional colours of the specific crownland they were in, for example in Bohemia it would flew together with the Bohemian white-red bicolour.

In Transleithania the Hungarian red-white-green horizontal tricolour was the one most commonly used (sometimes with the coat of arms). Again you could see it sometimes coupled with regional flags though this varied by region.

In both parts of the monarchy (aside from the land colours) you could also sometimes see nationalists flying their own flags but they weren't used to represent the whole monarchy, just their respective nations.

There was also a number of official flags, however they were not intended as national ones, like the flag of the merchant navy (often mistakenly considered as the 'national' flag of the Monarchy), the standard of the Emperor with the double-headed eagle or the war flag (Kriegsflagge).

Anthems:

The anthem had several several versions, the melody itself was composed by Joseph Haydn, with the text being prepared by Lorenz Haschka, and was first being played at Francis' birthday celebration in 1797. This version was the one that started with 'Gott erhalte Franz den Kaiser / Unser guten Kaiser Franz!'.

It served as his personal anthem until 1826 when the text was changed, the first stanza was left intact but the three other stanzas were changed substiantially, it stressed for example his commitment to 'German lands' as the Habsburg Monarchy was a leading member of the German confederation at that time as well as Francis' role in defeating the French and 'liberating' them. We do not know who edited this text however.

After Francis' death in 1835, Ferdinand became his successor. Since the text of the first two verses of the first stanza as well as the last two versers of each stanza would not make sense it was altered twice, first to 'Gott erhalte unsern Kaiser / Unsern Kaiser Ferdinand!' and in 1836 to 'Segen Östereichs hohem Sohne / Unserm Kaiser Ferdinand!'. The first change was made by a Prussian-born poet Karl von Holtei, the second one by Joseph Christian von Zedlitz, who came from Austrian/Bohemian Silesia.

After Ferdinand's abdication in 1848 and Francis Joseph's ascend to the throne, there was a bit of a 'chaotic' period where multiple versions of the anthem were being sung because there wasn't an official version for the new monarch. This lasted until 1853 when Johann Gabriel Seidl, a Viennese poet, was commissioned to write a new text to the anthem in order to have one before FJI's marriage to Elizabeth 'Sisi' of Bavaria. He did so and completed his work in 1854, the lyrics were nearly completely reworked, reflecting by the way the new political realities like the introduction of the concept of citizenship. This is the version that started with 'Gott erhalte, Gott beschütze / Unsern Kaiser, unser Land!' and being now named the 'Volkshymne'. This version was slightly edited following the rather unfortunate circumstances first by the birth of FJI's son Rudolf in 1858, then his death in 1889 and lastly the assassination of his wife Elizabeth (1898). During the somewhat brief reign of Charles I no additional changes were made.

It's also worth noting that it was translated from into different languages of the monarchy (with the lyrics being changed to varying degrees) and some of the nations within it had their own national (unofficial, of course) anthems.

source:

BARKER, Andrew, Setting the Tone: Austria's National Anthems from Haydn to Haider, in Austrian Studies, vol. 17, 2009, pp. 12-28