Paid contributions are ones deducted from your wages. Credited contributions are ones that you get usually when in receipt of a social welfare payment such as unemployment benefit or assistance. Both of them can count toward pension entitlements. There are certain classes of contributions that don't, but anything showing up as "reckonable contributions for pension" should count. So, for instance, the 2001 row on your statement is a full years worth of 52 reckonable contributions, even though it looks like only 38 of them are Class A contributions from employment.
The non-contributory pension is means tested, so if your mother only qualifies for that then yes, your father's income will significantly affect it. If she qualifies for a contributory pension it is not means tested. That is the main difference between the two, as the amounts are not otherwise vastly different.