As at least the foot component of the Royal Italian Guard was at Austerlitz I would like to know details about its composition. Usual OOBs note the Royal Italian Guard often only as Grenadiers but AFAIk there was a 1st battalion composed of 5 grenadier companies and a 2nd battalion composed of 5 companies of chasseurs.
So was there only the grenadier battalion at Austerlitz or both or were both combined into a single battalion?

Lachouque lists the ‘Brigade of Italian grenadiers’ on page 52 of The Anatomy of Glory. In 1805 there was one battalion of grenadiers of five companies and one battalion of chasseurs of five companies (both reduced from eight companies each in 1804). Together they formed the Guard Infantry Regiment and it appears that both served with the Grande Armee in 1805. Both battalions had Velites attached to them, which eventually became a Regiment of Velites which served in Russia. It was sometimes referred to as the Corps of Velites.

There was afaik a company of horse artillery from the Royal Italian Guard present, what was its exact designation in Fench?

The Italian Royal Guard had one company of horse artillery. It may have been in existence in 1805 and then may have served at Austerlitz. I have not found definitive evidence of either yet. The company's title in French was 'Artillerie a Cheval de la Garde Royale.'

Napoleon's escort came usually from the Chasseurs à Cheval de la Garde Impériale but had Austerlitz all squadrons clashed with the Russians. AFAIK Vélites were the escort for Napoleon but only the 8 companies of horse vélites or those of the foot too?

As the velites were mixed in with the Guardsmen on campaign, some were in the escort squadrons that served as Napoleon’s escort. When engaged in combat, the escort stayed with Napoleon and would not charge with the rest of the regiment. There were only two Guard cavalry regiments at Austerlitz, the Grenadiers and Chasseurs, so there would only have been four companies of velites, and these would have been assigned to the Guard squadrons and not serve together as complete units. Two companies of Velites were assigned to each grenadier battalion on campaign. There were 150 Velites in each Velite company. See L Fallou's La Garde Imperiale.

Saint Hilaire's work notes for 1805 2 battalions vélites grenadiers and 2 battalions vélites chasseurs each with 1,755 men. AFAIk the original composition was 1 battalion vélites grenadiers and 1 battalion vélites chasseurs each with 955 men.
So when was the number of battalions doubled and the how was the new company composition(because 1755 men does not match to a company size of 191 vélites)?

There were only one battalion each of velites in the Guard infantry regiments.

Assuming the foot vélites where not Napoleons escort, where were they and what did the do at Austerlitz?

The velites, both infantry and cavalry, were not fielded as battalions but were used to augment the Guard infantry battalions. Velites were added to the infantry and cavalry regiments in the Guard in 1804 and the cavalry and artillery got them in 1805. Each infantry regiment had a battalion of velites and each cavalry regiment a squadron (two companies).
Velites were not used as Napoleon’s escort as a unit.

Besides the 2 squadrons of Gendarmerie d'Élite de la Garde Impériale there was also a half battalion with 2 companies of foot gendarmes, where were they and what did the do at Austerlitz?
What was the exact designation in French for this battalion respectively these 2 companies?

The duties for the Gendarmerie d’Elite included the security of Napoleon’s ‘person, baggage, and residence’ and where he was living at the time, whether at the Tuileries or in the field. And they enforced his orders in the Grande Armee.

The two foot companies of gendarmes served at the imperial palaces and were ‘dropped’ in 1809. On campaign they would serve as security at Napoleon’s ‘palace.’

There were afaik 2 companies of horse artillery with guard, I hear sometimes the term squadrons for them, I assume the correct term is company, so are the correct designations for these units like this?
1ère Compagnie d'Artillerie à Cheval de la Garde Impériale
2ème Compagnie d'Artillerie à Cheval de la Garde Impériale

Two companies would be a squadron, both in cavalry and horse artillery. So there was one squadron of two horse artillery companies in the Guard in 1805.