One of the better sources on their status is the Final Report on the Progress of Demobilization of the Japanese Armed Forces. This is split into four PDFs, and the air forces are in part 3. I will quote extensively from this report, though I will go out of order.

First, the report includes these tables on the reported Japanese aircraft strength as of 1 October 1945. The values for early August were likely similar (they match closely with July 1945 estimates). I have expanded the shorthand labels for clarity:

Army Aircraft	Fighter	Bomber	Reconnaissance	Transport	Trainer	Other	Total
Japan Proper	2,268	489	641	147	1,951	155	5,651
Korea	187	45	147	14	266	251	910
Manchuria	188	9	27	8	810	0	1,042
China	62	2	9	0	57	0	130
Formosa (Taiwan)	145	17	17	0	9	0	188
Southeast Asia	0	0	0	0	0	920	920
Army Total	2,850	562	841	169	3,093	1,326	8,841

Navy Aircraft	Fighter	Bomber	Reconnaissance	Transport	Trainer	Other	Total
Japan Proper	1,618	1,183	226	0	3,343	704	7,074
Korea	1	1	0	0	43	7	52
Formosa (Taiwan)	85	44	9	0	202	60	400
Navy Total	1,704	1,228	235	0	3,588	771	7,526

Numbers of Navy aircraft in outlying areas had not yet come in yet, and there are November breakdowns by Home Island (with slightly different numbers, likely from early scrapping).

I would note that just because an aircraft is not listed as a combat aircraft does not mean it had no value. The trainers, for example, not only trained new pilots, but were also useful as kamikaze aircraft. On 31 August 1945 the US Navy published an Anti-Suicide Action Summary, essentially a guide on how to deal with such attacks for Operation Downfall, and I will quote from it on the last US destroyer lost in WWII:

During an attack on Task Group 95.5 at 0100 on 29 July the Japanese used biplane trainers. They were described as flimsily constructed of wood and fabric.

The action occurred during very bright moonlight, the enemy using several planes simultaneously in coordinated attacks. The Callaghan (DD792) was sunk and the Pritchett (DD561) was damaged.

Although the planes were believed to have approached at an altitude of 1,000 feet they were not picked up by any ship until they had closed to 13 miles. The maximum speed at which they were tracked was 90 knots, but they were extremely maneuverable and expertly piloted.

Use of such training planes in night attacks may be expected to increase. Despite their low speed these trainers represent AA. targets almost as difficult as the fast but less-maneuverable combat planes.

The threat was so significant the US used target drones of similar construction to see how effective radar and weapons were against such aircraft: in short, they were more difficult to spot and kill. Other weapons, like the MXY-7 Ohka (known as Baka at the time), presented their own challenges.

Aircraft production had fallen to under 1,500 a month by July (830 Army, 480 Navy), but the Japanese were dispersing as many facilities as possible, including moving engine factories underground (two noted made 50 each). They were also developing combat aircraft made from non-strategic materials. The Ki-106 was a wooden version of the excellent Ki-84, though I've never seen details on how the performance changed.

But numbers can be misleading:

Interrogation of high ranking officers and examination of pertinent documents during October developed a picture of the pre-surrender situation, which lent emphasis to the reasons for Japanese air-ineffectiveness. Figures indicate strongly that Japan was swayed toward surrender by the fact that her air force, though numerically substantial, was incapable of effective air combat resistance. This, in some measure, paralleled the European war in that after gaining of Allied air superiority, the enemy nation was destroyed without significant air opposition. The exception, however, in Japan's case is that surrender preceded ground invasion.

Early interrogation of high-ranking Japanese Army and Navy air officers, together with examination of Japanese air installations, confirmed previously deduced major factors which mitigated against the full use of this air strength. These principal factors were:

a. The dedication of all air strength to a suicidal short range defense of Japan Proper.

b. The distracting influence of a vigorous but tardy program to disperse aircraft and to transfer maintenance facilities, aircraft production, and hangars underground.

c. A serious shortage of aviation gasoline especially at strategic points, occasioned by our air-tight blockade of Southern Area resources which was crippling to training programs. Capability for flexible, rapid and coordinated air action was rapidly lost.

Inventories since have confirmed these factors, indicating clearly that the combat efficiency of the Japanese Air Forces was a hollow shell even before surrender. However, these forces would still have been capable of ultimate commitment to an all-out short-range and suicidal defense against actual landing forces.

After Okinawa in April and May 1945, the number of kamikaze and conventional attacks on US forces diminished, from 123 hits in April to 18 in June. The Anti-Suicide Summary notes:

To conserve his air strength for use in suicide attacks against our surface forces, and especially to prevent destruction of his aircraft on the ground, the enemy will make best use of concealment, camouflage, dispersion and deployment.

It is to be anticipated that in the Okinawa campaign the Japs learned they cannot seriously interfere with an amphibious operation by attacking picket stations [which were so heavily attacked I found one journal entry that calls it ""suicide row" for a very good reason"]. In future operations they may be expected to concentrate on troop and cargo transports and large amphibious craft before they have discharged personnel and equipment. Destruction of the enemy "at water's edge," a threat the Japanese heretofore have been unable to accomplish, will be attempted on a large scale in future operations.

Because of the vulnerability of air strips and their facilities to bombing, the enemy is concentrating on the production of float planes for use from Japanese rivers, lakes and harbors. These will be used chiefly at night, as will other suicide planes.

An intensive training program to prepare pilots for night operations is being carried on by the enemy.

This hoarding of resources was designed to preserve as much as possible for the invasion. There was no need to potentially lose valuable fighters in attacks on lone B-29s.

I don't have much information on numbers of aircraft available or the organizational state of the Japanese air forces in 1945, but I can answer at least one aspect of your question.

The reason B-29s were able to fly daylight raids unescorted was due to their ability to operate at altitudes that Japanese aircraft struggled to reach. Able to fly at 350 mph at around 30,000 feet, the B-29 was out of the reach of most older fighters and even all but the heaviest anti-aircraft artillery. Although the Japanese would get a handful of fighter aircraft into service by the end of the war that were capable of taking on Allied fighter aircraft on more even terms, they never really had anything in service that was capable of being a consistent threat to the B-29s at altitude. The problem came down to a few major issues - technological difficulties in engine development and material shortages. The technological issues were largely centered around engine development. Japan struggled to develop an effective supercharger for aircraft engines during the war - an issue that wasn't exclusive to Japan, but would have a major impact. Without an effective supercharger, the performance of piston engines dropped off massively with altitude, and at the high altitudes the B-29 was operating at, superchargers were absolutely necessary. Further impacting performance was the materials shortages that Japan was suffering by the time the B-29 raids had begun. The Allied blockade left critical strategic materials in short supply, including fuel and important metals for aircraft and engine production. Like Germany, Japan resorted to a variety of methods to stretch their dwindling supplies of fuel, and the general trend to the end of the war was that the quality of aviation fuel would decline. The impact of this is most easily seen in postwar tests of captured Japanese fighters - aircraft like the Ki-84 performed incredibly well in American tests with high quality fuel, but performance in Japanese service was significantly less impressive using Japanese wartime fuel. Similar issues with metallurgy impacted aircraft construction, making it harder to procure the metals needed for very high performance piston engines. Even aircraft structures were impacted by these shortages, as the regular failures of landing gear on the Ki-84.

So from the start, the B-29 was operating at altitudes where Japanese aircraft would struggle to reach them even in ideal conditions, and the declining quality of Japanese aviation through the end of the war left them increasingly unable to respond as the war came to an end.

Also by the fact that the Hiroshima bombers went unescorted seems to hint that the airforce was in a pretty poor shape by the time the war ended.

This was both because their air defenses were limited, but also because they did not want to draw attention to the plane. The US philosophy was that if the bomb was delivered by a single plane (with some observation aircraft not far away), the Japanese would not want to waste their precious aviation fuel chasing it down. By the late period of the war the US was bombing Japan in waves on a daily basis, and dozens of individual B-29s were used for weather and reconnaissance over the nation. Japan did not know an atomic bomb was coming, and so would not judge a single aircraft to be a major threat; even after Hiroshima, Japan lacked the ability to chase down every B-29 that entered its airspace.