Pretty late to reply but here are the answers to most of your questions:
Q1:How did they manage to add Bandar abbas and Jawadar into their Nation?
A1:After a rivalry between the two sons of the last Sultan in the Ya'ruba dynasty the Persians managed to Invade Oman under the rule of Nadir Shah,after many rebellions one Man called Ahmad ibn Sayyid Busaidi revolted and successfully kicked the Persians out, he would go on to liberate Basra and Bahrain from Persian rule on his reign,After building a navy more successful than the Persian one he rented the Persian ports and overtime it became under Omani rule. (He also installed the Busaidi Dynasty which rules Oman to this day.)
https://en.wikipedia.org/wiki/1717_Omani_invasion_of_Bahrain
more about this here:
History of the Oman and Zanzibar Sultanate
Q2:How did they have colonies in East Africa?
A2:The Omanis were colonized by the Portuguese in the early 16th century, they never really kicked them out until the 17th century, at that time Portugal had Colonies all over East Africa, so their locations were known to the Omani Sultans, The Omani Navy/Army captured many forts along the African coast with the help of African tribes and thus kicked the Portuguese out of the Horn of Africa.
https://en.wikipedia.org/wiki/Siege_of_Fort_Jesus
OMANI ARAB RULE IN ZANZIBAR
HISTORY OF ZANZIBAR
Q3:How strong was their Navy/Army?
A3:It was decent, in the Indian Ocean the navy was only second to the British, the Millitary was good enough to Defeat the Persians on many battles and siege/defeat the Portuguese in East Africa,in terms of actual numbers the Omani Navy in the 19th Century had about 100 Warships outfitted with 30-74 Cannons and a couple hundred small Trading Ships (for the Trade in India,Africa).
The Millitary equipment was basic,An Omani Matchlock "Abu Fathilah" and a Sword and a Rhino Shield
Omani soldier Guillain 1845
I'd say they had 12,000-20,000 Soldiers and 7,000 to 11,000 Reserves.
Q4:Any famous Sailors/Naval Leaders?
A4:Actually yes, Since the European Traders were looking for a way to get to India, They asked Arabian Sailors to guide them once they got to Africa/Arabia,One of the most famous Omani Sailors was
https://en.wikipedia.org/wiki/Ahmad_ibn_Mājid
Ahmad bin Majid was born on Modern Day UAE, which at the time when he was born it was Oman, the UAE never really got independence until the British came and Made it a seperate entity, before that Oman and the UAE had the same history, Same nation etc.
Q5:How were the Relations with other Nations?
A5:Nice, they formed relations with many nations such as the USA,India,Persia,U.K,France etc.
USA:American traders wanted to get in the African trade so they negotiated with Oman which after the Omani ship "Sultanah" Visited the Capital of the US, Relations were good.
France:Omani ships visited Southern France and an Embassy was formed in Oman by the French in the 19th Century.
I hope this answers most of your questions.