Gentile was a neo-idealist, whose philosophical position gradually evolved into what he referred to as "actualism" (there is another philsophical position which goes under the same name, but is unrelated). As a neo-idealist, Gentile embraced a tradition firmly critical of Marxism, especially of economical and political Marxism, (Spaventa, Croce, and among others Pareto, Böhm-Bawerk) and was just as critical of the Italian socialist movement, especially after the "maximalist" turn of 1919-20.

It is probably outside the scope of this answer to illustrate in detail Gentile's philosophical position. But, in short, he wasn't a socialist, he didn't believe in a "strong government" (mixing up government and State would be inconcievable to a neo-idealist, and especially to one who grew to identify the State with the actualized spirit of the nation - in late-idealist and neo-idealist terms, the State would be the "real which is rational", the spirit the "rational which is real" and the government the magmatic expression of passion and afflictions of the individual and collective mind which explained the "irrational" whirlwinds which, at times, perturb the course of the historical process), and - this is a bit more tricky due to Gentile's own difficulties in reconciling his early criticism of the Italian Nationalists with his support for the Fascist Regime which had integrated many of those nationalist positions - would not have framed it as "love for your country".

While he did alter his position in this regard, one thing which remained consistent is the fact that Gentile never regarded the Nation (not the Country, which is just the territorial expression of it) as a "given" external to the "nation-people". The Nation existed in so far as the collective spirit "actualized" it with a constant act of volition, and its characters therefore, far from deterministically and materialistically determined, could change - or better - could be brought to coincide with the "reason" inherent to all collective beings.

This is, it goes without saying, a very modest attempt at sketching Gentile's philosophical position in relation to the points you made. It is far from exhaustive, and only summarily accurate.

As to Gentile being a "father" of Fascism, the father of Fascism was - in Gentile's own assessment - Mussolini. Indeed, Gentile's aligment with the growing fascist movement only begun when the philosopher started to recognize in the new movement certain elements of the process I sketched above (late 1921 - early 1922). This was the time of his first, controversial, declarations of support for the fascist violence; where he explained that (anti-socialist) violence was always a "moral act". Or that, in other words, the act of violence which brought reason and rational order was "ethical", even when it wasn't practiced by the recognized authorities.

From there on, Gentile became one of the most prominent intellectual figures of the Fascist Regime, but his main ideological contributions (the so called Dottrina del Fascismo) came significantly later than the Fascist affirmation and even more so when one considers the original formation and definition of a fascist ideology. Those were, in good measure, divulgative works rather than academic ones, as Gentile was also busy with his various Ministerial and Directorial functions, which echoed similar "popular histories" as those produced by Gioacchino Volpe, and aspired to retell a narrative of fascism as continuation of the Italian Risorgimento, rescuing the weak slumbering liberal Italy by means of the "radiant days of May" and thanks to the Italian intervention, the victorious coronation of the Great War and finally the "Fascist Revolution" of October 1922. Along these lines the ideal spirit of fascism had both caused the fall of the inept liberal governments, and offered its strength and will to restore the authority of the State threatened by the violent assaults of the "Italian Bolsheviks".

Which brings us to the last point. When did Fascism became associated with the right? It was - matter of fact - when they begun assaulting and burning socialist organizations, newspapers, leagues, threatening socialist figures out of their workplaces, hometowns. Forcing the dissolution of socialist administrations and imposing their replacement with suitable "national" candidates.

Among agrarian workers, fascist squads found a prominent position in breaking-up strikes, and subsequently in destroying the foundation of the socialist presence, both economically (cooperatives, leagues, unions) and politically. The opposition between Fascism and Socialism was a discriminating trait of the Italian political landscape already by mid 1920; but already in 1919 many political groups of the so called "democratic interventionism" had expressed their doubts on an electoral list including, or led by Mussolini, due to their fear of losing the "discontent socialist" vote, as the latter would not have voted for a man who was perceived, albeit a former socialist leader himself, as a champion of "bourgeois reaction".

For general reference:

Alatri, P. - D'Annunzio, Nitti e la questione Adriatica

Albertini, L. - Vent'anni di vita politica

Arfè, G. - Storia del socialismo italiano

Colarizi, S. - Dopoguerra e fascismo in Puglia (1919-26)

De Felice, R. - Mussolini – vol. 1, vol. 2

Di Scala, S.E. ; Gentile, E. - Mussolini 1883-1915

Forsyth, D. - The Crisis of Liberal Italy

Furiozzi, G. B. - Il sindacalismo rivoluzionario italiano

Gentile, E. - Le origini dell'ideologia fascista

Gentile, E. - Il mito dello stato nuovo, dall'antigiolittismo al fascismo

Gentile, E. - Storia del partito fascista: 1919-22, movimento e milizia

Malagodi, O. - Conversazioni

Melograni, P. - Storia politica della Grande Guerra

Noiret, S. - Campagne elettorali e sistemi elettorali nell'Italia liberale 1900-1924

Noiret, S. - La riforma elettorale del 1918-19

Noiret, S. - Riformisti e massimalisti in lotta per il controllo del PSI; 1917-18

Rochat, G. - L'esercito Italiano da Vittorio Veneto a Mussolini

Rochat, G. - L'Italia nella prima guerra mondiale

Rochat, G. ; Isnenghi, M. - La grande guerra

Roveri, A. - Aspetti della lotta politica dal 1919 al 1926; in Storia dell'Emilia Romagna

Roveri, A. - Considerazioni sul consenso al regime fascista; in Storia dell'Emilia Romagna

Sacchetti, G. - Il Sindacato Ferrovieri Italiani dalle origini al Fascismo, 1907-25

Vivarelli, R. - Il fallimento del liberalismo

Vivarelli, R. - Storia delle origini del Fascismo

On the matter of Hitler 'considering himself on the left', by no means. We've got an FAQ section on this specific question (How Socialist Was National Socialism, if your browser doesn't take you there immediately), with answers from u/kieslowskifan, u/g0dwinslawyer, and u/sergey_romanov.