Also, it is worth firing of a quick email to ask if they have registered anything, and specifically, what they have registered. I wouldn't enter into any other correspondence until that information were given.
Hi, it is decided on a case by case basis - basically a judge has to consider the two together to determine that, but there is general guidance (albeit really, little more than logic) here: http://www.ipo.gov.uk/types/tm/t-other/t-infringe.htm. As you can see from his language in his email, the person who wrote to you has clearly read this or something similar.
There is not enough info here to look at the specifics of the issue, although I do note that you are seeking general advice as to the line you should take - in short what degree of pragmatism you should exercise.
That is a difficult one to answer. You mention lack of time. It is always difficult to ascertain at the start of a problem how much time an issue will take, simply because it is impossible to determine how far the other party are willing to push things. If they dig their heels in, the time expenditure could be considerable. If they are flying a flag, they may simply go away after an email or two - the problem is, you just never know.
In terms of costs, again, it could be a simple matter and cost a few hundred pounds and be resolved by a few letters. They may dig their heels in and you then face either spending more, or finding a work around. Again it comes down to whether they are simply flying a flag, or whether they are prepared to push matters.
Realistically, I would suggest you need to ascertain your legal position in terms of:
a) Whether they do have a trademark;
b) If so, whether your name/logo is close enough to breach that;
I would try to speak with them - they may offer an initial free appointment for a start up, and take specific advice on your problem, then decide the way you want to go.