Aun así, sigo ilusionado con el viaje del verano y ya estoy mirando hoteles.

In this fixed expression, aun así means even so / nevertheless. Here aun usually has no accent because it means even, not still/yet. A good test is that it is close to incluso así.
You may sometimes see aún así in real life, but the usual standard recommendation is aun así.

Seguir + adjective means to continue being in that state. So sigo ilusionado means I’m still excited / I’m still looking forward to it.
If you said estoy ilusionado, that would simply mean I’m excited, without the extra idea that this feeling has continued from before.

Yes — especially in Spain, ilusionado often means excited, really looking forward to something, or full of enthusiasm. It is a false friend if you read it as English illusioned.
So estoy ilusionado con el viaje means I’m excited about the trip, not I’m under an illusion about the trip.

With ilusionado, Spanish often uses con to introduce the thing you are excited about. So ilusionado con el viaje is the natural way to say excited about the trip.
This is just a common Spanish pattern: adjective + preposition. It does not always match English word-for-word.

In Spain, mirar hoteles is very natural for looking at / checking / browsing hotel options. It often sounds a bit more casual, like you are viewing possibilities.
Buscar hoteles would sound more like searching for hotels with a more goal-focused feeling. Both can work, but mirar is very common in everyday Spain Spanish for this kind of browsing.

Not necessarily. Mirar hoteles usually means looking at hotel options, not actually reserving one yet.
If they were definitely booking, you might expect something like reservando hoteles or haciendo una reserva. Here it sounds more like the speaker is researching or comparing places.

Spanish often leaves out subject pronouns because the verb ending already shows the subject. Sigo and estoy clearly tell you the subject is I.
So (Yo) sigo ilusionado... is possible, but yo is unnecessary unless you want emphasis or contrast.

Y simply links the two ideas: the speaker is still excited, and as a result is already looking at hotels.
Ya adds the idea that this has already begun. It helps show progression: not only is the speaker excited, but they have already started taking action.