When someone with a car gives you a ride somewhere you want to go, you should not offer to pay half the gas, you should offer to pay *all* the gas. They are providing the vehicle usage, the depreciation due to mileage accrued, the pro-rated maintenance costs, the insurance, the registration, etc. Those fees easily match the price of the gas, plus they are doing you a favor since you don't have the car on your own. You should offer to pay all the gas and be happy about it. I don't think I've ever in my life seen other people do this correctly. Often the passenger makes no offer at all because they think "you were going there anyway, I don't add any cost". If they're polite they will offer to split the expenses somehow. No.
Asking someone to drive you to the airport is very rude and completely inappropriate. A taxi is very cheap and convenient, or there are plenty of other means. Getting someone to drive you takes at least an hour of their time, which costs $100-$200 or more. But that's the least of it. It ties them into doing something for you at an exact hour, which has an even greater hidden time cost because they can't do anything that will intersect that time period. Plus, if your flight is delayed or something, then their appointment is delayed, which ties them into all the annoyance and vagaries of your flight. Furthermore, if you fuck up and make yourself late or schedule it wrong or something, they inheret all your stress from that. It's an absolutely massive thing to ask of someone. It's pretty inappropriate even if you're completely broke. If you are broke and need a ride to the airport, the more considerate thing would be to ask a rich friend for cab fare (though the rich friend is probably too damn stupid to realize you're doing them a favor and agree).
When cars "let someone in" who's trying to merge or turn in or something, they think they're being so nice, they feel all fucking considerate and superior. Not so, it highly depends on the situation. In general by doing that you are making a decision whether to inconvenience the person who wants to get in, or the person behind you. First of all it's pretty fucking egotistical to decide that you are the person who gets to decide whether the person behind you or the person trying to merge gets your favor. Secondly, more often than not, you should favor the person behind you. For example, if you are on a major arterial and there are lots of cars behind you and some guy is trying to pull in, you should NOT let him in, because doing so severely slows the flow for many people behind you. The utility of each action is multiplied by the number of people affected. Obviously there are cases where you should let someone in, like if they've done a dumb thing and got themselves stuck where they're blocking traffic going the other way; in that case not letting them in is a huge dick move. Like most things in life, the correct action is dictated by consideration of the alternatives and how they affect various people, which is what people never do.
On a related note, this is somewhat more interesting of a topic, I think it's a bad idea to help the traffic flow of people who choose to take a very high congestion route. Let's say there's a certain commute pattern in a city that has very high congestion. In order to reduce it, the city makes it easier for those people to fan out onto alternate routes to commute (by providing travel time information for detours, or doing a little bit of through street connection). This reduces their commute time and most people are happy. I contend that this is a bad thing. You have hurt the traffic flow of the people who were on those alternate routes and avoiding the bad commute path. I contend that you actually want the bad pattern to be very painful, because it encourages people to not do it. Similarly you want the good patterns to be as pleasant as possible, because those people should be rewarded for avoiding the bad pattern. Traffic has a sort of self-regulating property where if you reduce it, more people drive and it gets back to equal badness again. I contend you generally don't want to do anything for it, and especially you don't want to spread it out onto alternate routes. The thing I'd like to see more is bypasses and better local routes so that people who are avoiding the bad traffic paths aren't affected by them.