Can I stay in the US even if my I-94 expires?

This is very risky. Read the instructions of the change of status form at uscis.gov. If you marry a U.S. citizen, you can stay even if you had overstayed.

Your body belongs to your own but any other people. Therefore, you can stay anywhere you want. However, if your concern is whether you may be in a position to change your non-immigrant status, then it is imperative that you maintain a valid status before applying for such change. If you are not clear about this written advice, then it is time for you consult a lawyer in person and make sure you pay for his/her service!!!

If you would file the change of status before the expiration of your I-94, you may do that.

You can remain in the US as long as a change of status application is filed prior to the I-94 expiration. However, if the change of status is not approved, you are considered out of status going back to the expiration date of the I-94.

If you are asked at the POE, you should tell them that you will be visiting few colleges. When you decide on a college, have I-20, then you can think about changing status. Having dual intent before you enter the U.S. may prevent you from changing your status in the future.

No. You become a non-immigrant visa "Overstay" if you remain in the US even one day past the date granted to you by the expiration date on your current B-2 I-94. This will cause you significant problems in the future if you are classified as an "Overstay". You should not plan to stay past the date you were authorized. Additionally, you will have a difficult time in most cases since 2001 requesting a change of status from B-2 tourist to F-1 student. Your best procedure would be to visit schools while you are still in valid B-2 tourist status and then depart the US prior to the date granted by the admitting officer at your POE. Once you have received an acceptance letter from the school you wish to attend in the US you should then set up a consular appointment in your home country and process accordingly for the issuance of your F-1 visa. DHS will no longer allow a person in B-1/B-2 status to begin school until s/he has changed status to F-1 or obtained an F-1 visa abroad and re-entered. [8 C.F.R. 214.2(b)(7)].

You can apply for a change of status from B2 to F1 as long as you submit the necessary forms and paperwork before your authorized stay expires, even if it expires after the application has been submitted. You should not mention about the change of status until you actually find a school and start the process.