Hi Jenna,
In relation to the client searching their Business name, that shouldn´t be a big issue (unless they have chosen highly competitive generic words for their business name like "low interest loans") as EGOL mentioned they should rank fairly fast for this.
For adwords you can show them this http://www.tbkcreative.com/why-you-shouldnt-google-your-own-ads/ . Many times though the clients will still continue to search anyway. One of my clients only stopped after i showed him in his account that the keyword he was constantly searching had a ctr of 4% where all others in the account were more close to 20% and that it was the only one with a Quality score 3 points lower than the rest. After this i showed him the results of him having stopped searching and he hasn´t restarted. In this case though we were talking about 15-20 searches a day minimum. Sporadich searches shouldn´t impact that much.
You need to also setup a regular reporting where you can show the results and this should be part of the inicial agreement. Here you can show either data from analytics, GSC, Adwords, Moz etc... that will assure them there are actual results from your work. And be very clear from the start about what they can expect.