The answers to both of your questions (1. Does shemitat kesafim annul the loan, or merely forbid the lender from claiming? and 2. Does shemitat kesafim happen automatically?) would appear to be the subjects of arguments among the Rishonim.
Many Rishonim would appear to understand that shemitat kesafim is indeed a full, automatic annulment of the loan.
However, Yere'im seems to be of the opinion that the loan continues to exist, but the lender is forbidden to claim it, and needs to explicitly state his intention to do so in order to free the borrower of his obligation to repay.
Either way, the practical result is as summed up by Rambam in Hilchot Shemitah v'Yovel 9:28:
כָּל הַמַּחֲזִיר חוֹב שֶׁעָבְרָה עָלָיו שְׁבִיעִית רוּחַ חֲכָמִים נוֹחָה הֵימֶנּוּ. וְצָרִיךְ הַמַּלְוֶה לוֹמַר לַמַּחֲזִיר מַשְׁמִיט אֲנִי וּכְבָר נִפְטַרְתָּ מִמֶּנִּי. אָמַר לוֹ אַף עַל פִּי כֵן רְצוֹנִי שֶׁתְּקַבֵּל יְקַבֵּל מִמֶּנּוּ.
Whoever returns a debt after shemitah, the Rabbis are pleased with him. The lender needs to say to the borrower, "I cancel it, and you are under no obligation to me". If he [the borrower] says, "Nevertheless, I wish for you to accept it", he [the lender] accepts it from him.
For more details and quotes, see below the line.
1. Does shemitat kesafim annul the loan, or merely forbid the lender from claiming?
The simple implication from Rambam Hilchot Shemitah v'Yovel 9:4 is that the loan is annulled.
וּכְשֶׁתִּשְׁקַע חַמָּה בְּלֵילֵי רֹאשׁ הַשָּׁנָה שֶׁל מוֹצָאֵי שְׁבִיעִית אָבַד הַחוֹב:
When the sun sets on the evening of Rosh HaShanah following shemitah, the debt is lost.
This is further indicated by his ruling in 9:28 that if a borrower nonetheless wishes to returns a loan after shemitah, he should stress that he is doing so as a gift, rather than as a repayment of any loan that continues to exist.
וְאַל יֹאמַר לוֹ בְּחוֹבִי אֲנִי נוֹתֵן לְךָ אֶלָּא יֹאמַר לוֹ שֶׁלִּי הֵם וּבְמַתָּנָה אֲנִי נוֹתֵן לְךָ:
He should not say, "I am giving it to you to repay my debt", rather he should say, "It is mine, and I am giving it to you as a gift."
However, Yere'im 278 (in the old edition) argues that in fact, the debt does continue to exist, and it is simply prohibited for the lender to claim it.
נראה לי דשמיטה דקרא אינה מחילה, אלא ציווה הקב"ה שישמיט, פירוש: יניח, ולא יתבענו עד שיתן לו מעצמו, דכתיב 'שמוט' ו'לא יגוש', שכל 'שמיטה' הכתוב בקרא הנחה ולא מחילה, דכתיב 'והשביעית תשמטנה ונטשתה', פירוש: תניח. הלכך, הלוואה לעולם מוטלת על הלווה, שלא יעכב על זה חובו לעולם בתיבתו לאוצרו, ואם עשה כן, הוה ליה 'לוה רשע'
It seems to me the שמיטה in the verse is not 'forgiving', rather G-d commanded שישמיט, meaning, 'he should leave' and not claim it, until he [the borrower] gives it on his own. As it states, "Leave" and "Do not press". Because שמיטה always means 'leaving' and not 'forgiving', as it is written, "And in the seventh year leave it and abandon it", meaning "leave it". Therefore, the loan continues to exist as a debt against the borrower, and he should not keep this debt forever in his closet or treasury, and if he does this, he is a 'wicked borrower'.
Note that even in Rambam's view, although the debt is totally annulled and there is no obligation on the borrower to return, if he nevertheless chooses to do so (as a gift) the Sages are pleased with him.
Note further, that even in Yereim's view, that the debt still technically exists, the borrower will never be forced by a court to repay; rather the court will force the lender to renounce his claims - see the end of part 2 for details.
2. Does shemitat kesafim happen automatically?
Mordechai, Gittin 527 (quoting R. Avigdor HaCohen) argues that it does take place automatically.
לענין השמטת כספים קבלתי, שהשביעית משמטת מאליה, אע"פ שלא אמר 'משמט אני', אלא מצוה הוא דרמיא עליה באמירה, דכתיב 'וזה דבר השמטה'... זכר לדבר: נולד לו בכור בעדרו מצוה להקדישו, אע"פ שכבר קדוש מאליו משעת לידה
With regards to shemitat kesafim I have received a tradition that the shemitah year cancels loans automatically, even if the lender does not say "I cancel". Rather, there is a commandment for him to state it, as is written, "This is the matter [lit. word] of the shemitah" ... This is similar to the case of a firstborn animal being born in one's flock, where there is a commandment to sanctify it [verbally], even though it is automatically sanctified from the time of its birth.
However, Yere'im 164 (in the new editions) argues, claiming that nothing happens without the lender's declaration.
וחוב שעבר עליו שביעית, אינו רשאי לוה לעכבו אלא על פי מלוה, שכל זמן שלא השמיטו מלוה, חייב לפרוע, אלא לוה יזמין מלוה לדין, שישמיט לו חובו כאשר צוה היוצר, ובית דין יחייבו למלוה לומר 'משמיט אני' ... ואם אינו רוצה המלוה לומר 'משמיט אני', יכפוהו בית דין
Regarding a debt after the shemitah year, the borrower may not keep it without the say-so of the lender. Until the lender declares it to be annulled, the borrower is obligated to repay it. Rather, the borrower should invite the lender to court for the purposes of cancelling the loan as the Creator commanded, and the court will oblige the lender to say "I cancel" ... And if the lender does not want to say "I cancel" the court will force him to.
(It seems to me that this is consistent with his position described above in part 1; if the loan continues to exist, then the borrower needs to make efforts to repay it, unless and until the lender has formally stated that he does not intend to press for collection of the debt.)