I feel nervous because I don't know if it's okay or normal for him to invite me over to his home in such an occasion
Don't be nervous, Poles invite people from outside their family to Christmas Eve supper and I'm sure your friend asked his parents/the hosts whether he can invite you (of course, it is a family thing and the people who are being invited probably aren't just any people, so you may feel honoured by such invitation, I guess :)).
The only awkward, confusing moment for you may be during breaking the opłatek
If your friend's family is celebrating this tradition then it will probably happen at the beginning. Everyone will get one Christmas wafer and they will be breaking them with each other and exchanging wishes (while standing) before the supper. If you get one then people will come up to you (one person at a time) and wish you stuff like good health and fortune, etc. and while they're giving their wishes you should break off a bit or a few bits of their Christmas wafer (they will be holding it in front of you). And then (at least if you know some Polish, I guess :)) it will be your turn to give your wishes and they will be breaking off bits of your Christmas wafer (so offer it to them like they did with their wafers). Btw, take only white wafers, those blue, pink or yellow ones are for animals (if the family has pets). You usually eat those bits of wafer after you finish the breaking of the wafer with one person and then you move along to another person.
If giving wishes is over and you're still left with some wafer then eat it or put it on your plate (don't throw it away into the dustbin or sth).
The breaking of the Christmas wafer isn't a religious ceremony or anything, it's more of a family tradition nowadays but the wafers are baked in the same way as Hostia (sacramental bread used during Mass) and breaking of Christmas wafers has its roots in old Christian traditions so if you have a problem with taking part in it you can simply tell your friend beforehand that you'd like to be excluded from this.
If the family is more religious someone may read a fragment from the Bible and after that the eating starts and that's basically it :) According to the tradition there should be 12 dishes and in theory everyone should try at least a bit of each dish. But that's just theory - I usually don't eat each and every dish, mostly only those I like ;) (you can ask your friend which are the most edible ones, or if you want to be more polite - which ones are his favourite :)) If the host asks you about the food then it would be probably better to say that everything is very tasty ;)
Btw, in my family we start the Christmas Eve supper when the first star shows on the sky (if it isn't cloudy, that is :)).
Do not openly drink alcohol on the 24th
I haven't met this custom.
It probably depends on the family.
Be polite, eat at least a little of everything offered to you at the table, compliment them on the cooking,
As for gifts, a box of very fine chocolates or a selection of very fine teas, perhaps an excellent bottle of dessert wine, fancy liqueur or mead (miód pitny) for mom
Another option could be one of those hand painted Christmas bulbs (usually a bulb holder is included), I guess. But I don't think it's really necessary to give gifts to other members of the family (except for the hosts).