I need some advice from you. I decided to start exercising with weights
besides doing my power walk. We have a basic choice of equipment and I plan to use only 5 machines that exercise different body parts. So, the questions are as follows:

1. How many days of the week should I use the weights? I plan on 3 or
4

3 or 4 are ideal. Try 3 and make it four if the wife is nagging you... ;)

2. How many repeats on the same machine (I do 10 reps)

About 8-12 repetitions if you want to gain strengh. 12-20 would be more
aerobic and are good for an exercise such as sit-ups.

3. How many times to repeat the whole process (I do it 3 times)

About 2-4 sets of any type of exercise. How many types of exercise? Depends
on how much time you are willing to spend. 24 sets are quite a standard.

4. I also like doing sit ups on a slant board. Is this worthwhile?

Surely. Just make sure you don't lean back fully flat on the board. Keep
your back bent forward at all times and concentrate on the range of motion
that is defined by the point where your back is parallel to the ground and
about 45 degrees w.r.t. the floor.

5. When to increase the pounds of weight? Just go by what feels right,
or
increase after a certain time period?

When you are able to perform 8-12 reptition easily. If you exceed that
number, you know the increase is recommended.

6. What length of time do you recommend I spend working out? I am
planning on doing about ½ hour.

30-40 minutes for about 20 sets is quite standard. Don't work out
excessively. Listen to your body and be sure to know when it wears out and
adjust the duration of your workot accordingly.