I have to make two decisions regarding whether I will choose to spend time with Christians.

The first is with my Mother whom I live with, as she has planned a leisure trip tomorrow. The area that we may most likely go is by the water of a lake, where people will likely be wearing very little clothes. I also am not sure whether she will want to go on a boat ride where there will almost certainly be music playing. If I do not go, she will likely go by herself. The drive will be quite long as well. While she is my mother, it feels as though she has taken on many other roles in my life being that she received minimal assistance from my family in raising me and my brothers. Could this have any effect on increasing the rights granted to her through fiqh? And will it be bad akhlaq to decline the invitation?

Another situation I have is that a former friend who I met before I became Muslim, who has asked me to meet with him and a mutual female aquaintance. I am not quite sure if this meeting will be of any real benefit to anyone. From the perspective of fiqh and akhlaq, would this be recommended if there is no guarantee that my mind will not float into thoughts of lust?

Islam has greatly emphasized on the rights of parents especially the rights and a mother, whether she be a Muslim or non Muslim.
It is only at the expence of disobedience towards Allah's commandments that a child is allowed to disobey his or her parents and that too should be done with utmost politeness.
One should always keep their moral behaviour and show warmth and kindness towards them even if they're harsh. Such a policy can change the outlook of a parent about their child even if they seem to disregard each others views generally speaking.
You are definitely facing a predicament on how to spend your time with two people who dont share the same faith like you.
If accompanying both of these people will lead to sin and you can stop that by excusing yourself politely, then it is incumbent that you do so.
If however your mother will take it to heart and feel much remorse, then you can explain her first about why you don't want to go, and if it were not for the music and other forbidden sins, then you would certainly accompany her in the trip.
If she still feels very bad about this that you feel it will completely spoil your relationship in the long run, then go along with her but at the same time try all you can do avoid sinning. You can do zikr and isteghfar ie remember Allah and ask for forgiveness when the music is heard, and lower your gaze when you see any one who is ill dressed and not Mahram ie lawful for you.
As for your friends, you can simply excuse yourself if you have to take your mother, and if not, you will still have to think of.anything but to let go off the probable awaited sin of falling into lust.

Remember they don't have any right as such over you and all the more so since they are non Muslims.
For humanity sake, you can still show your good behaviour and keep your relations with them but never at the expence of your faith.

Remember, the Almighty Lord is always besides you when you do the right thing.