Again, just adding my own. Great answers so far:
I’m used to: Enter church, praise, offering (and greeting others), worship, sermon, benediction, dismissal. The video I watched of a service seemed to have (I don’t really know how to put this) a lot of chanting. What happens with regards to the actual sermon?
The sermon (or homily) is usually given after the Gospel reading, and before the Liturgy of the Eucharist (or "of the Faithful") but also can be given at the very end of the Liturgy. You're probably used to a long 45-minute sermon...not so for us! Homilies half that length are still considered quite long. My priest tends to have fifteen minute homilies, and he's mentioned that our bishop has tasked him with cutting that down to seven!
I understand that there’s no “confession” per se, as in the Roman Catholic sense. To this end, what do you do if you have a problem and you need to speak to someone, ie the Father? Is it possible to make an appointment to see them?
We do have sacramental confession as in the Roman Church (though our theology about sin and atonement is different that the RCC or Protestant perspective, but I'm sure you'll hear more on that later). Traditions differ about how often this is done. Usually it's between once a month or quarter-annually. Though, sometimes it can be only once a year (before Pascha...not widely practiced anymore) or every week before Liturgy. Generally confession is offered after the Saturday evening service (either Great Vespers or Vigil), which begins the celebration of the Lord's Day (Sunday). Appointments can also be made for confession and a host of other things. It'd be just fine for you to make an appoint to just chat!
I understand that kissing is used as a sign of reverence and respect (for example, that’s why you kiss the priest’s hand). However, what about when it comes to things like the kiss of peace – what do you do if that is something that you don’t feel comfortable with doing?
I regret that I've never been to a parish in the U.S. that actually does the kiss of peace. I think that's a shame, it's a beautiful custom! But, for a lot of folks, it's offensive to American sensibilities of personal space. I don't think you'll offend anyone by offering a handshake instead.
What is the significance/meaning of the hand gesture that Jesus is making in the icon?
Secondly, I understand this is from inside the Hagia Sophia, and so it makes sense to depict the (current at the time?) rulers. Why are they so close to Jesus though? From what I can gather, neither are saints. If I'm not mistaken it's Constantine IX and Empress Zoe.
These icons both show Christ making the "christogram", which spells out "IC XC" an abbreviation of "Jesus Christ" in Greek (Iecouc Xrictoc). This should help:
Unfortunately, I don't know much about the iconography of the Agia Sophia. I know it is common to depict rulers in icons, especially if they funded the project (a "benefactor's icon"). I regret not having much information about this particular mosaic, though. Sorry.
It is my understanding that when you become baptized (chrismated?) and formally enter the Church, you are given a new name, often related to that of a saint. How do you go about choosing this name, and is it “complete”, ie, you have to get a legal name change in your country, or is it something that fellow members of the Church will call you?
Just to clarify, if one is baptized, then they are also chrismated. Some can be only chrismated, having a previous baptism considered "valid" enough to not warrant being baptized by the Orthodox. Different bishops will choose differently on how to receive converts in this matter. but, personally, my original baptism was in a Baptist church, and I was received into Orthodoxy by baptism.
Secondly, you may
receive a new name. You will receive a patron saint. For example, my legal name is Benjamin. I'm named after my grandfather. When I was received, I was baptized as "Benjamin" and took St. Benjamin the Deacon, a 5th century Persian martyr, as my patron. My name didn't change. I know a lot of priests that encourage converts to keep their names, if there is a saint with that name. Though, I know plenty who took a different name than their legal name as well. I don't know anyone who has legally changed their name. Most will go by their Christian name only at church...some only in a liturgical context (e.g., that's what you're called when you're absolved, when you commune, etc. anytime you're mentioned in services) but never in conversation, even with other church folks. Of course, this whole mess is resolved if you just take a patron with the name you already have...if that's an option for you!
I hope that was helpful for you. And, of course, welcome to the Boards!