My experience is that for the most part all transistors need to come out of the circuit to get an accurate read on their condition; a good variable heat iron really comes in handy for this task. Yes, you can catch the shorted and the open transistors in circuit but the leaky junctions really need to be isolated to test. I use a B&K 520B, a Sencore TF151A, and the fellow in figure 2.7 of this page:
http://www.tpub.com/neets/book21/87d.htm I use a couple of 1K resistors in the circuit instead of the listed resistors. The procedure is to check the base/emitter junction, check the base/collector junction and then check emitter/collector (which should be open.) My signal generator can put out about 12volts p-p so you get a graphical view of what the junctions are doing from -6 to +6 volts. I used a very similar circuit to check transistors in RF equipment years ago when I was doing bench work full time. This will catch problem transistors that my other two testers will miss.
As far as replacement transistors go generally you need to get a datasheet on the old transistor and line up the specifications of a prospective replacement to be as good or better than what's coming out. In the case of power transistors things like type (PNP or NPN) case (T03, T066 etc), power dissipation, max voltage base to emitter, collector to emitter, and so fourth. You may be able to find replacement recommendations on line or in a cross reference. Bipolar transistors for the most part really have fewer variables than vacuum tubes.