your agent is the one to answer your questions, from what I know
1.母亲还没有拿到LTSV,请问现在如何才能拿到LTSV?是需要密集多次来新加坡呢,还是需要单次呆很长时间?
ans: to apply LTSVP, you mother has to be in Sg. after arrival, you may apply for LTSVP. the application will take 4-6 weeks per ICA website and what you can do is to apply online, make necessary payments and wait for approval.
2.现在公民还能和LTSV的父母买祖屋吗?
ans: Yes you can.
只有母亲一人拿LTSV可以买吗?
ans: Yes.
这样买的祖屋是一手的还是二手的呢?
ans: 2nd hand resale HDB.
对公民的收入有要求吗?
ans: There is said to have a salary ceiling above which the person is not allowed to buy HDB. You may ask your agent for details.