Is it considered a Sunnah to take a different route when going to and returning from the obligatory prayer, and is the intention to go to a distant mosque out of a desire for good deeds considered a Sunnah?

It is from the Sunnah for one who performs the Eid prayer to return by a different route than the one taken to go, and this is established from the guidance of the Prophet, peace and blessings be upon him, and it has rulings and benefits. Scholars have differed on whether this Sunnah extends to the Friday prayer and the five daily prayers. Some have gone to the extent of deeming it desirable in them, while Sheikh Ibn 'Uthaymeen preferred that it is specific to the two Eids only, citing that the Prophet, peace and blessings be upon him, did not do so in other than Eid, despite the presence of a reason.

As for going to a further mosque out of desire for reward, there is nothing reported to indicate its Sunnah, and Sheikh Ibn 'Uthaymeen preferred that it is better to pray in the nearest mosque, unless it is in Makkah or Madinah, then it is better to go to the Grand Mosque or the Prophet's Mosque even if they are far.