"the value of training the core muscles? Yes/No? Why? Performance benefit? Technique benefit? Relation with injury prevention?"
Core strength being important isn't the question, because it is. The real question is it the MOST important? It depends on how weak or strong your core is, compared to your ability to swim, bike and run. Specificity is most important, (swimming, biking and running), but there comes a point where you can't improve those if the weakest link in the chain is your core strength. Also, how much training time do you have available? That's one of the best parts about triathlon, that there are so many different avenues for improving performance, that choosing which ones you focus on and how well you do with that is the challenge.
So does it have a benefit on the things you list? Yes, most certainly. Is it worth pursuing more? Perhaps, but see if it is the really the weakness you need to address. With all this said, I do like to see my athletes address core strength mostly in the off-season, then maintain that strength thru the year, but just enough to maintain.
"have u blogged about if & How much warmup you should do the morning of the race & How long before race time should u do it"
Warm-up in general has a few rules I like to follow and advise of athletes. First, the shorter the race, the more intense it is, and therefore the more warm-up that is required. For example, a sprint or Olympic requires much more warm-up than an Ironman. As an example, I like to see an athlete do at least a 2 mile jog before a half-ironman, and then get in the water for at least 20 mins, with some surges in both the run and the swim, to race intensity. This would be the minimum for a half. Always better to start your warm-up too soon than too late.
"how bout approach to offseason? need to incorporate weights to gain strength. how much sbr do we need not to lose form? Thanks!"
The first part of approach to the off-season is to recharge mentally. If you don't rest the mind, you'll be fading by the middle of the following season. I believe strength training is best addressed in the off-season, with a focus on stabilizing muscles, and core work. Don't fall for the idea that a loss of strength will mean poor technique, as that is just not the case. Plenty of weak athletes have excellent technique.
The best approach to the off-season is one which represents the glaring weaknesses that kept you from reaching your goals in the prior season. Do a review of the season, and find out what happened. If it was injuries, perhaps a strength training routine could prevent that. Maybe speed is what you need too, and that needs to be the focus. Consider all variables, then prioritize. Consider weight and diet too!
"What type of strength training do you recommend for marathoners?"
Wow, strength training is a popular topic! Seems like many athletes haven't addressed it much, and that may be why it seems to be a big weakness. Because a runner only has to run, they tend to be more able to hit the weight room, so strength training comes into play. Core work, stabilizing muscles, and dynamic movement strength work, (ie lunges), are the things I recommend.