No. Life expectancy estimates are based on current information alongside extrapolation of trends
I can't speak for how every country does this, but the basic methods are common. I'm summarizing the figures used by the UK's Office of National Statistics (ONS).
Current projections for the UK population are:
Around one-third of babies born in 2013 in the UK are expected to survive to celebrate their 100th birthday.
Of the 374,000 men and 393,000 women aged 65 this year (2013), 8% and 14% respectively are expected to survive to their 100th birthdays according to the principal projection..
Both these claims are much more pessimistic than the first two made in the question. One third of babies making to 100 is not a "reasonable expectation" of living to 125 for newborns nor is 14% making it to 100 for 65 year olds.
So how do they calculate these numbers?
Life expectancy is calculated by observing current mortality (that is how many people in each age group actually died last year) and extrapolating trends in those rates into the future. The key information is those mortality numbers which are current as most countries are fairly diligent about recording deaths. We can't predict the future perfectly, though, so the extrapolation to future mortality will introduce some uncertainty (but the projections are founded on long series of clearly observed numbers so, again, are not based on "old" data).
The ONS explains their method in more detail thus:
A life table is purely a hypothetical calculation. It is a statistical tool typically used to portray expectation of life at various ages. The basic assumption is that a given number of births (100,000 born in a given year, known as the radix of the life table) are subject, as the survivors pass through each year of age, to the mortality rates prevailing for each age.
Period life tables deal with mortality rates in a particular year only. The mortality rates for each age are used to calculate how many of the 100,000 births will reach each year of age until eventually all have died. This enables the total number of years lived to be calculated. When this total is divided by the number of persons in the life table (100,000), the result is the average number of years lived or the mean expectation of life at birth. The total number of years lived from any given age can also be calculated and when divided by the number of survivors entering that year of age, the figure obtained is the expectation of life in years for those persons.
Cohort life tables are calculated using age-specific mortality rates which allow for known and projected changes in mortality. A cohort life table provides mortality rates that vary over time for each age. For example, cohort life expectancy at age 65 in 2012 would be worked out using the mortality rate for age 65 in 2012, for age 66 in 2013 for age 67 in 2014 and so on. This uses observed mortality rates in 2012 and projected mortality rates from 2013. Therefore, cohort figures are regarded as a more appropriate measure of how long a person of a given age would be expected to live on average than period life expectancy.
There are two projections being made here. Period life expectancy uses current mortality in its estimate; Cohort life expectancy uses projected future mortality changes (so it is the one to compare to the question's claims).
For the UK the latest numbers are shown in the ONS chart below (with three scenarios based on uncertainty in the future mortality numbers):
Life expectancy is calculated based on current, observable, numbers not old assumptions. And it isn't reasonable given those numbers that the optimistic projections in the question are likely to occur.