BTW I am mostly speaking about early islamic history ie; rashidun, umayyad and abbasid caliphates

Question 1: I'm really curious about whether Christians and other Peoples of the Book can fight for an Islamic leader under a non-forced circumstance?

Simply yes, Non-Muslims were allowed to fight for a Muslim sovereign (whether that be a sultan, caliph, emir etc.). And there were very few cases of non-Muslims being forced into serving Islamic armies, in fact it was almost the exact opposite, Non-Muslims were not drafted into the army, instead they paid the jizya tax because they were dhimmi or “protected”. But non-Muslims were allowed to join the army and fight alongside Muslims. These non-Muslim soldiers were seen as fighting for Islam, and as such these people were exempt from paying the jizya.

What if I'm a devout Orthodox Greco-Syrian in the living in Damascus in the 12th-13th centuries, and I'm widely known as the best fighter and commander in the region? Could I openly fight for the Sultan or Emir and my loyalty is never questioned?

I believe you are talking about two things, would you be able to be a champion, and would you be able to be a commander, so I will answer those. Firstly I have no sources for non-muslim champions in muslim armies, thus I believe they weren’t used because it would send the wrong sort of message, a non-muslim is the best of the muslim army.

But there have been a few rare cases where non-Muslims have commanded Muslim armies such as Samuel hanagid, but usually non-Muslims were not allowed to lead armies of Muslim soldiers. There are cases where non-Muslim princes* of Muslim empires have led large contingents of Muslim soldiers, but that is because it was their levy.

Also about perceived loyalty, non-Muslim soldiers did not usually hold high positions so the loyalty of these people was not a major concern.

*Not really princes see Mansabdars

Hopefully answers your question.

Ottomans for sure had some Christian soldiers in their lines.

While not probably exactly what you want to know, in Ottoman borderlands and in general newly conquered lands plenty of soldiers, neither majority nor the core of garrisons, but mostly auxiliaries called martolos, were christians.

In addition to this troops, rich Sipahis who owned larger timars would also need to bring additional troops with them, and, especially when speaking about lands in Europe it wasn't that uncommon that some of those would be Christians. Actually you can even find some sipahis holding timars in 15 and 16th century who were themselves christians still (basically keeping their lands as reward for defecting to ottoman side). However those were incredibly rare, and many such christians would very quickly convert to Islam in order to participate and advance in social sphere.

However as the cavalry arm (sipahi) was based on land-possesion, Christians would be rare. You would be sooner to find Chrisitans in some specialized troops like cannoneers or firearms infantry.

Ottomans constantly kept an emphasis on the firearms, and while they had plenty of their own Muslim troops, the Ottomans were very eager to hire anyone who had experience and skill in this, not hesitating to hire Christians. To give some examples:

Gabor Agoston shows from the records of various Balkan forts that in the 15th century Ottoman gunners and cannoneers and arqebusers were in large numbers christians. The ratio was smaller in larger forts, and through time changed even more in favor Muslims, but still some Christians always remained

Even foreigners would be admitted. C.F. Finkel shares a case of garrison of several hundred French/Waloon muskteers situated in the Hungarian (Habsburg) town called Papa during the Long War (1593-1600). Because they were being unpaid, they defected to the Ottomans in 1600. They fought on the Ottoman side for the remainder of the war and we even have records of them being kept on Sultan's salary in the aftermath. Some of the men converted to Islam, but not all, nor was it specifically demanded from them. Throughout the same war, there are documented cases of Ottoman troops hiring Polish, Transylvanian and even 'Austrian' troops in certain campaigns.

Sources:

Gabor Agoston: Ottoman Artillery and European Military Technology in the Fifteenth to Seventeenth Centuries

Gabor Agoston: Firearms and Military Adaptation The Ottomans and the European Military Revolution, 1450-1800

C.F. Finkel: French Mercenaries in the Habsburg-Ottoman War of 1593-1606: The Desertion of the Papa Garrison to the Ottomans in 1600

Ive Mažuran: (In Croatian) Povijest Hrvatske od 15. do 18. stoljeća