The Taiping had very little impact on the Perry Expedition, but the Perry Expedition had somewhat more on the Taiping; the Taiping had some impact on the Boshin War, but that happened after the Taiping War concluded.
1: The Perry Expedition
It is no coincidence that the two American attempts to send diplomatic missions to the Taiping capital at Nanjing came before and after Perry's period of expeditioning in July 1853-April 1854. There was in fact a major rivalry between Perry on the one hand, and the American consuls in China on the other, one which ended up having significant ramifications for American policy in China.
Colonel Humphrey Marshall, who was dispatched as consul in August 1852 and arrived in early 1853, differed from Perry on two counts. Firstly, Marshall saw China, not Japan, as the critical East Asian power, and the one where American interests should lie. To quote one of his letters to Perry, dated 13 May 1853, he claimed that 'There is nothing to be hoped for in Japan equal to the advantages now actually enjoyed in China.' Secondly, Marshall believed that as a consul and thus a diplomatic representative, he had the right to appropriate American naval resources as needed to secure what he saw as American interests in East Asia. This included Perry's flagship, USS Susquehanna. In early March 1853, during the interim between the recall of Commodore John H. Aulick and the arrival of Perry at Hong Kong on 9 April, Marshall commandeered the Susquehanna and sailed to Shanghai. Perry's letter back to Washington, written the day of his arrival, barely concealed his irritation:
Whatever may have been the urgent necessity of sending away a ship of the squadron at a moment when my arrival must have been hourly expected to assume the command, and when I should have cheerfully co-operated with Mr. Marshall, it would I think, have been more judicious of Commander Kelly to have taken Mr. Marshall on board his own ship to Shanghai or have sent the Saratoga, rather than have removed from my control for an uncertain time, the steamer assigned as my flagship.
Marshall's first attempt to reach Nanjing in early April 1853 ended rather ignominiously: the ship was damaged when it struck some rocks near Zhenjiang and was briefly abandoned by her crew. But Perry's arrival did not lead Marshall to reduce his efforts, rather the opposite. Now, he demanded a ship to take him to Tianjin, where he intended to press the Qing court to allow US consular staff to be permanently resident at Beijing. Perry, owing to the fact that he would need to assign yet another ship to Marshall, and also, it seems, due to his own objection to what he saw as needlessly aggressive policymaking on Marshall's part, refused to allow this, and while he left a ship in Shanghai it was to be there as a reserve, ready to sail out to Japan if needed.
However, the US government actually supported Marshall over Perry, and when the first official response from the State Department to Marshall arrived in September, he was given instructions to the effect that he was allowed to appropriate whatever naval resources possible to ensure the security of American citizens and interests in China during the ongoing crisis. However, Perry refused Marshall's orders to dispatch ships from Shanghai and Canton to the lesser treaty ports (Ningbo, Fuzhou, Amoy) or to provide naval support to the Qing, and insisted on maintaining a fleet for a second Japan expedition. One key factor that played into this was that Perry was strongly pro-Taiping, which Marshall very much was not. From Perry:
...I must be the judge of the necessity of using the force at my disposal in intermeddling in a civil war between a despotic government struggling for its very existence... and an organized revolutionary army gallantly fighting for a more liberal enlightened religious and political position; and hence my undeviating policy, whatever have been my sympathies for the revolutionists, of practicing myself, and enjoining all under my command, a studied regard to neutrality and non-interference.'
Would it surprise you to learn that Perry was a Northerner, while Marshall was from Kentucky and went on to join the Confederacy?
Perry's delaying tactics succeeded. In mid-January 1854, as part of a reshuffle of foreign consuls under Franklin Pierce, Marshall left China, having been replaced by Robert McLane (later a Democratic politician in his home state of Maryland), but McLane carried with him further State Department instructions indicating that Perry should allow any resources superfluous to the Japan missions to be used in China. Nevertheless, Perry refused to detach the Susquehanna from Japan until his mission concluded, such that McLane was only able to set out to the Taiping capital at Nanjing in May.
While there, relations between the Americans and the Taiping were distinctly terse. At Taiping-held Zhenjiang, warning shots had been fired at the ship, something taken as a grave insult by her captain, Franklin Buchanan (another Southerner, who went on to command the ironclad CSS Virginia), and the Americans repeatedly refused to clarify the reasons for their visit. Again, ignominy marked the end of the Susquehanna's. second visit, as eight crewmen trespassed onto restricted land, one even attempting to climb the Porcelain Tower, and were arrested and interrogated before being released, along with a message stating that the safety of further trespassers would not be guaranteed. McLane's subsequent reports were not keen on the Taiping's religion, nor optimistic about their chances of victory.
So, how did the two affairs play into each other? Perry's stubbornness meant that the consuls really had very little ability to influence the Japan expeditions. Communication times with the US were simply far too long to be of much use: Marshall's first complaint about Perry went out in late April 1853, and by the time he received a response in late September, the first expedition had already been to Japan and back. But it also meant that any significant American activity in China was curtailed until the Japan expedition ended, quite possibly hardening the American stance on the Taiping despite, ironically enough, Perry's pro-Taiping leanings. While Marshall evidently was not outright pro-Taiping at the start (his intention to strengthen US-Qing ties in April 1853 is evidence enough), he may well have been more open-minded, and given access to the desired resources (namely a shallower-draft steamer) he might have arrived at Nanjing with potential for more cordial relations. In the event, the 13-month delay meant more time Marshall (and latterly McLane) spent communicating exclusively with Qing authorities and already-jaded British and French consuls, further pushing the tenor of reports back to the State Department towards an anti-Taiping bent, and in turn creating a more aggressive American policy intended to extract more from the Qing (whom they expected to win the war), rather than support the Taiping. While the US would not be an official belligerent in the Anglo-French war with China known as the Arrow War or the Second Opium War, small numbers of US forces were involved, and the United States would come to be a signatory to the treaties of 1858 and 1860 that concluded the war's two main phases, which in turn thus created an economic stake in ensuring Taiping defeat.
Much of the above section is based on Chester A. Bain's 1951 article, 'Commodore Matthew Perry, Humphrey Marshall, and the Taiping Rebellion' in The Far Eastern Quarterly Vol. 10, no. 3, though the parts about the Susquehanna's successful voyage are based on Jonathan Spence's God's Chinese Son.