You mix many different issues in one. Nevertheless:
1). Substitution reactions are always in competition with elimination ones. Therefore, substitution reactions are often accompagnied by elimination side products
2). Primary halides undergo Sn2 and tertiary halides undergo Sn1. But secondary halides may undergo both mechanisms and sometimes simultaneously, depending on case.
3). Hydrolysis of 2-bromopropionic acid in basic medium, occurs by Sn2 mechanism, regardless the neighboring of the carboxylic group. E2 is not very favored in aqueous medium
4). Hydrolysis of 2-bromopropionic acid in acidic medium, occurs mainly by Sn2 mechanism. Sn1 leads to a carbocation that favors E1 elimination and obtaining of acrylic acid that is stabilized by conjugation, before hydrolysis occurs.
5). Double dehydrohalogenation would lead to formation of alkynes or allenes that are very difficult to obtain due to thermodynamic reasons. Probably, dehydrohalogenation will stop to the vinyl bromide derivative. Except, if the double dehydrohalogenation is favored by specific conditions, e.g. neighboring of carboxylic groups or an enol formation that leads to a conjugated carbonyl. Contrary and according the literature, double dehalogenation is feasible with Zn and vic- dihalides.
6). I hope that such really complicated issues that apparently seem simple, will not be asked at the exam.