The first question is really unanswerable because of a simple fact: ships move. A number of major and minor ships served in both theaters; USS Iowa for example initially served in the Atlantic Fleet in case the Tirpitz tried to break out and later transferred to the Pacific in January 1944 where she served for the rest of the war. RN ships could also have quite a long operational theater. HMS Prince of Wales had the misfortune of not only facing the Bismarck while working up, but also was part of the ill-fated Force Z that defended Singapore a few month later. The unique needs of both theaters (technically, three-Atlantic, Pacific, and Mediterranean) meant some ship types were more common than others and a few ships stayed entirely within the same theater for the conflict. There was very little need for the heavy American carriers or their cruiser escorts in the Atlantic, so ships like USS Enterprise stayed mostly in Pacific waters. RN carriers and cruisers were more than capable of dealing with the small German surface fleet and the hobbled Italian Navy. Smaller support craft like destroyers and logistical ships also tended to remain in the same theater as it made little sense to move these vessels around the globe due to fuel costs. But getting a firm percentage of Allied shipping per theater is impossible because the numbers were always in a state of flux.

Leyte is a bit easier to answer.

One of the salient things to realize about Japanese naval planning post-Philippine Sea was that no Japanese plan or gambit was going to work over the long-term. The Mariana's Turkey Shoot had shown that American naval forces not only had quantitative superiority over the Japanese, but qualitative as well. The scale of the task before the Japanese was impossible. But as John Prados has cogently pointed out in a recent study of the Leyte campaign, just because Japanese strategic goals were unreachable does not mean that IJN officers planned poorly. What would become the Sho plan was an attempt to realistically accomplish the impossible.

The IJN chiefs felt that they had two big assets after the loss of three carriers and the bulk of their carrier pilots in the Marianas: a heavy surface fleet and a still strong land-based air establishment. Of the two, airpower needed the most attention. IJN land-based airpower had taken a beating in the Marianas, but the American offensive was approaching some of the inner imperial core of Japan. This meant that both IJA and IJN possessed an number of pre-existing air assets that could be used. There were plans for joint Army-Navy operations such as T-force, a special mixed unit to attack in bad weather, and the IJN also sought to encourage the IJA to think more proactively about attacking naval targets. The Turkey Shoot had also underscored the efficacy of American radar and fighter control, so Japanese air attacks started to take this into account using terrain to mask approaches and using chaff to denude radar of its effectiveness. But these efforts to rectify land-based airpower could not rectify the deep problems in aircrew training as well as the yawning technological gap between American and Japanese aircraft. The IJN did envision a resuccitated carrier striking force built around Katsuragi-class carriers and the converted Shinano, but even the optimists estimated that carrier-qualified air groups would only be available in the spring of 1945. In this cold logic, the IJN knew it could treat its existing carriers as expendable assets and they became part of a strategic feint in Sho.

The problems of adjusting the surface fleet to new realities were more surmountable. Although Sho envisioned land-based airpower as a striking force, its Japanese planners recognized that continuous air cover was not possible. Therefore the IJN surface fleet received a massive beefing up of its AA defenses. Some of these involved new weapons like AA rockets or the Akizuki-class AA destroyer with the new Type 98 gun, but most of this effort involved packing the surface fleet with proven AA equipment like the 25 mm Type 96 and the 127 mm Type 89 DP gun. Over the course of the summer and autumn of 1944, the IJN added approximately 2000 Type 96s to its various surface ships and converted cruisers like the Isuzu and Maya into specialty AA cruisers with more AA weapons and enhanced fire-controls.

Japanese planners also had the benefit of accurately assessing the American strategic objective of the Philippines. A 25 July communique by the Japanese Naval General Staff warned on the basis of American staff meetings in Hawaii that the next offensive would be in the Philippines and much of the Sho plans were predicated on defeating a Philippines invasion. Other Sho plans also considered thrusts against Taiwan or the Home Islands. The basic idea behind Sho was to concentrate the best aspects of Japan's remaining striking force and defeat an American invasion or significantly blunt its naval power. IJN estimates of its own losses tended to be fatalistic and some of Sho incorporated abortive plans to relieve Saipan with a one-way mission of obsolete battleships that would use surprise and determination to swing the battle in Japan's favor. IJN planners like Ugaki Maotome did not feel that piecemeal or pinprick attacks on the US would accomplish much given the growing numbers of American ships and planes. IJN intelligence may not have had the advantages of American cryptoanalysis, but it did appreciate the massive numbers arrayed against Japan. The IJN instead would divide its surface forces to scatter American material superiority which would then have to contend with the IJN surface ships suddenly converging onto the main American force.

But the execution of Sho befell the Robert Burns's famous aphorism of mice and men. American carrier raids on Taiwan in early October 1944 triggered a premature execution of the Sho plan for Taiwan by the local commander Kusaka Ryunosuke. The only Japanese forces in the region, the land-based air units, met the American carrier attacks and predictably failed. Although IJN aircraft did take advantage of lessons learned in the Philippine Sea, these lessons were not enough. Kusaka aptly referred to the efforts of his airmen to being like throwing eggs against a wall. Thus the Sho plan for the eventual Philippines was denied its air-based component outside of local forces in the Philippines. Additionally, Sho's underlying concept of using disjointed forces acting to converge together looked fine on paper, but was much harder to execute in reality. To Sho's credit, the disjointed IJN surface groups did add an element of confusion among American admirals during the Leyte campaign as they could not correctly ascertain where the main IJN force was during these attacks. IJN AA as well as ASW proved to be inadequate for the task at hand and Kurita's force lost two heavy cruisers to submarines before the campaign began and its AA defenses could extract higher punishment from American air attacks, but not stop them.

The biggest problem with Sho was that no matter how intelligently the IJN planned to accomplish its goals with the tools at hand, the Japanese were always pursuing a strategic dead-end. Inflicting a sharp naval defeat on the US in the Philippines might have stopped the invasion and this would have been a huge boost to morale in the Home Islands, but it was putting off the inevitable. The reasons for Kurita's indecisiveness at Samar is one of the vexing issues in the historiography of Leyte ( this previous answer of mine goes into some of the details), but some of the debate misses the important aspect of Sho that victory was an end onto itself in Sho. Exactly what the IJN won was largely immaterial. In Third Reich historiography there is a German phrase that aptly fits the IJN's situation: flucht nach vorn. Roughly meaning "fly forwards," the idiom means when in doubt, double down and attack while disregarding more logical responses. Much of the IJN's planning fits within this paradigm. Ugaki's diary entries post-Philippines show an increasing preoccupation with meeting the immediate tasks before him and deliberately eschewing the larger strategic picture. The hazy strategic ends of Sho enabled Kurita's Samar turn precisely because the overall plan did not designate any concrete objectives for the Japanese naval efforts.

Sources

Peattie, Mark. Sunburst: The Rise Of Japanese Naval Air Power, 1909-1941. Annapolis: Naval Institute Press, 2002.

Prados, John. Storm Over Leyte: The Philippine Invasion and the Destruction of the Japanese Navy. New York: New American Library, 2016.

Tully, Anthony P. Battle of Surigao Strait. Indianapolis: Indiana Univ Press, 2014.