I've had some experience in running a training program for faculty advisors and working with new advsiors at my institution in the US (you haven't said so, but I'm guessing that you're at a US institution.) I've been asked a lot of very similar questions in recent years. Your questions show that you care about doing a good job!
Your questions also seem to show that your university isn't providing enough support to faculty advisors. I'd argue that having incorrect information on the web site and pushing new programs for undergraduates without providing relevant information to the advisors are signs of disfunction in the advising system at the university level. You may be able to work around these issues, but a better long term solution is to encourage the administration to fix the system.
At a very low level, you can address systematic problems like this by being persistent but polite in reporting issues to the responsible authorities. For example, emailing the registrar's office to tell them that information on one of their web pages is contradicted by the current catalog should probably result in the incorrect web page being updated. Don't complain about these things in a very public way though. Some administrators welcome this kind of assistance while others dislike "interference" and might ignore you. If you run into that attitude, then your best approach may be to escalate to that person's boss.
At a higher level, issues like these can be addressed through faculty committees. If you have a faculty committee that deals with advising issues (or more generally with academic procedures) then you can bring these issues to the attention of the committee. Don't be surprised if you're recruited to work on the committee!
Here are some suggestions of things that you can do on your own to be a better advisor:
Learn as much as you can about the academic rules and keep up with changes. In working with students, either be absolutely certain of the answer to a question or know who to ask to get an authoritative answer to that question. The worst mistake that you can make is giving incorrect information to a student. In comparison, "I'm not sure of the answer to this question" is much better.
Make sure that you have correct information about any student that you're advising. Know how to use your Student Information System to look up a student's transcript, run a degree audit, etc. Many advising mistakes are the result of advisors not having complete and correct information about their advisees.
If a student is in danger of "flunking out" due to bad grades, it is important to intervene as early as possible. If the problems are academic rather than personal than you should work with the student to come up with a realistic plan to address the student's bad grades. Make sure that you know all of the rules about academic probation/suspension and that you're aware of the options that a student might have for withdrawing from classes. Monitor your advisees' performance by checking their midterm and final grades each semester. If your campus has an early alert system then respond to flags raised on your advisees.
Crisis management is important. When a student who has been doing well suddenly starts doing poorly, this can very rapidly result in loss of financial aid, academic suspension, etc. Often times these situations are the result of personal problems (health problems, psychiatric illness, drug abuse, death in the family, breakup with a boyfriend/girlfriend, etc.) These kinds of personal problems are not things that you should address, but you should refer the student to offices that can help (a campus health center, counseling office, etc.) In these situations it is often best for the student to cut back on their academic commitments by dropping courses or even withdrawing from an entire semester.
Academic decisions can affect financial aid, and the rules are so complicated that faculty advisors generally aren't in a good position to answer questions about them. Check with your financial aid office on the consequences of any action that a student takes that changes their number of registered credit hours.
Advising students should ideally be more than just course selection. The traditional "prescriptive advising" approach of simply telling students what courses to take should be replaced by a "developmental advising" approach in which the advisor's job is to help the student understand the choices available to them, clarify their goals, and then take responsibility for their educational path. This kind of advising can be more rewarding for both students and advisors but it takes a lot of work. Developmental advising can be particularly helpful for those students "of moderate ability and motivation" by helping them to figure out what they want to do in life and having that serve as motivation to do the hard work necessary to be more successful students.
The reality is that a BS/BA in mathematics is not by itself a qualification that is likely to lead to an exciting career in mathematics. Most students are going to need to combine this with either a second major or a more advanced degree. For example, there are lots of opportunities for students with MS degrees in applied math, statistics, OR, etc. Some web sites that do provide useful information on careers for students majoring in mathematics include:
You should get to know the "career services" office on your campus. I encourage my undergraduate advisees to attend job fairs and look for internship opporunities starting very early in their undergraduate career. Finding out that many employers have GPA cutoffs at 3.0 (or even 3.5) can have a powerful impact on a student who has been getting lower grades...