but I don't know if I should bow my head,
This is meant to show respect, symbolically indicating bowing to God. If you don't see a reason to show such respect to God, then I don't see why you should so bow.
That being said, I don't recommend raising your nose (without also looking upward towards Heaven), or any other sort of disrespectful action, if your goal is not to communicate disrespect.
close my eyes,
As far as I've been able to determine, this is really more an issue of tradition than a religiously backed stance. In John 11:41, the text mentioned that "Jesus lifted up his eyes" before praying (just before a miracle). Also, John 17 verse one starts out describe "Jesus" ... "lifting up his eyes to heaven", praying.
or say "Amen!"
Please don't. The word "Amen" is not fully understood by most people who say it. (They think it is just meant for something indicating the close of a prayer, perhaps like "Good bye" to God.) However, the word is sometimes called out during a church service, indicating agreement.
I've been taught the word means "I agree". Based on usage, I think some other phrasings may also make sense, like "Agreed", or "That's truth". It is an affirmative word.
So, if you don't agree with "God, we thank you...", then it would be better not to say such a word, as the common consensus would be that such lying/dishonest speech would be violating one of the Ten Commandments.
These aren't events where I'd normally expect a religious component; we're typically at someone's house having dinner, and not in a church, or at a wedding or other ceremony.
It is good to see people treating religion seriously, rather than just trying to limit its impact to only affect one day a week (when church is held). I'm not quite sure why you would expect religious people to not have religion be a part of their home.
Many people have been brought up with "saying grace", referring to praying before eating. I suspect that may be based on the religious text of Daniel 6:10 referring to praying three times a day, and that many people have 3 meals a day.
This sounds pretty normal. By that, I'm simply saying that I believe a lot of people do that.
I generally hold hands with the people around me (even though this makes me uncomfortable)
I've experienced this. I believe this is more about causing a feeling of unity among the participants, and is not a practice that has any specific heavily religious meaning.
I also feel conspicuous and uncomfortable by not participating fully (although I feel far more uncomfortable just by being asked to participate).
Somehow, my mother ended up not teaching me some simple childhood prayers that were known by children from multiple other households from her side of the family. They would be recited by people whose eyes were wide open, so they realized I wasn't saying the words. I never did spend quite enough time with them to completely learn those words. I simply observed without participating in the speech, which I was really unable to do. As far as I know, nobody ever expressed/demonstrated/showed any judgment against my behavior.
I do want to respect the people around me, however.
How should I behave to be the most respectful of the people around me?
There isn't just one universal answer that will definitely work in all circumstances, as some people will have different expectations. Whatever I tell you may run the risk of one of your family members taking offense to whatever you decide.
Of course, the approach that would generally be preferred by the religious people is to join 'em. (That is, to become a believer.)
If that's not going to happen, then I think that simply avoiding religious activities (like praying) can be quite sensible. For instance, if you walk into a room after you hear the group "Amen", that doesn't seem overly disrespectful, in my opinion.
If you decide to join the group, such as sitting around the table (perhaps simply because that is logistically most sensible as you participate in a family event), then you would show people plenty of respect by simply staying quiet, allowing the speaker(s) to speak their prayer, and allowing all others to listen. Naturally, this includes not making silly faces for youngsters, or riling them up in some way to create a disturbance. Silencing your cell phone is also a great idea.
If it helps, just think of the national anthem at a sporting event, or watching multiple uniformed professionals synchronously shoot guns at a college graduation (yes, I've seen this) or a funeral. Simply watching, and not being disruptive for anyone who is wishing to have this be a moving time, is sufficient respect.
As discussed in this question, I am fine with being politely silent and still during the prayer.
What that question doesn't cover, however, is to what degree is it respectful for me to participate or not participate?
Take comfort in realizing that you're showing the most respect that can be reasonably expected from someone who doesn't fully embrace the beliefs behind the activity.